Structures of Critical Nontree Graphs with Cutwidth Four

Abstract

The cutwidth of a graph G is the smallest integer k ($k\ge 1$) such that the vertices of G are arranged in a linear layout $[v_1,v_2,...,v_n]$, in such a way that for each $i=1,2,...,n-1$, there are at most k edges with one endpoint in $\{v_1,v_2,...,v_i\}$ and the other in $\{v_{i+1},...,v_n\}$. The cutwidth problem for G is to determine the cutwidth k of G. A graph G with cutwidth k is k-cutwidth critical if every proper subgraph of G has a cutwidth less than k and G is homeomorphically minimal. In this paper, except five irregular graphs, other 4-cutwidth critical graphs were resonably classified into two classes, which are graph class with a central vertex $v_0$, and graph class with a central cycle $C_q$ of length $q\le 6$, respectively, and any member of two graph classes can skillfuly achieve a subgraph decomposition $\mathcal{S}$ with cardinality 2, 3 or 4, where each member of $\mathcal{S}$ is either a 2-cutwith graph or a 3-cutwidth graph.

1. Introduction

The graphs under consideration in this paper are finite, simple and connected, and for the undefined graph-theoretic terminologies, we refer the reader to the book by Bondy and Murty [1]. The cutwidth of a graph G is the smallest integer k ($k\ge 1$), such that the vertices of G are arranged in a linear layout $[v_1,v_2,...,v_n]$, in such a way that for each $i=1,2,...,n-1$, there are at most k edges with one endpoint in $\{v_1,v_2,...,v_i\}$ and the other in $\{v_{i+1},...,v_n\}$. The method used to compute the optimum cutwidth of a graph G is usually referred to as the cutwidth minimization problem, and has received an enormous amount of interest in graph theory literature [2] since the 1950s. From [3,4,5,6], for a graph G and a nonnegative integer k, deciding whether the cutwidth value of graph G is less than k is an NP-complete problem for general graphs except for trees, and it remains to be $NP$-complete even though G is planar with a maximum vertex degree of 3, by [7]. Therefore, most of previous investigations of the cutwidth problem have been mainly concentrated on polynomial time approximation algorithms for general graphs, and on polynomial time algorithms for special graphs for solving their cutwidth [2,4,5]. Despite these theoretical algorithms of the cutwidth minimization problem, research on studying the structural properties of the extreme (or critical) graph classes whose cutwidth is a given integer value $k>1$ have been paid little attention. As far as we know, the 2-cutwidth graph class has five forbidden subgraphs ${\tau }_1$–${\tau }_5$ [8] (see Figure 1 below), the family of 3-cutwidth trees possesses 18 forbidden subtrees [9], and 50 forbidden subgraphs of unicyclic graphs with cutwidth 3 were also found by [10]. As for the inner structures of the critical graphs with cutwidth k, ref. [11] found that any critical tree with cutwidth value k can be decomposed into three $(k-1)$-cutwidth subtrees which are either edge-joint or edge-disjoint. Recently, the decomposability of a class of special k-cutwidth critical graphs with a central vertex $v_0$ and at least two cut edges $v_0{v}_1$ and $v_0{v}_2$ was also characterized by [12]. However, for general critical graphs with cutwidth $k\ge 4$, their inner structural properties are unfortunately not yet known. The cutwidth minimization problem for graphs has many significant applications. In the early 1970s, Adolphson and Hu used it to model the number of channels in the optimum layout of a circuit [13]. Other applications of this problem include $VLSI$ circuits’ layout [14,15], automatic graph drawing [16], network reliability [17], information retrieval [18], urban drainage network design [19] and others. In particular, the cutwidth is closely connected to a basic parameter called the congestion, in designing microchip circuits and micro communication element system [2,20,21]. Herein, a graph G is considered to be a mathematical model of the wiring diagram of an electronic circuit, in which the vertices of G mean components and the edges of G represent wires connecting these vertices. When a circuit is embedded into a certain architecture (say, a path $P_n$ or a cycle $C_n$), the largest number of overlapping wires is referred to as the congestion, which is one of the key parameters determining the electronic performance. These are of great interest to scholars investigating the cutwidth problem in graph theory practically. Theoretically, the cutwidth problem is also closely bound up with other graph parameters such as bandwidth, modified bandwidth, pathwidth and treewidth [2,22,23]. For example, this is the case for any graph G with vertices of a degree bound by an integer $r\ge 1$, $pw\left(G\right)\le c\left(G\right)\le r·pw\left(G\right)$, where $c\left(G\right)$ and $pw\left(G\right)$ are cutwidth value and pathwidth value, respectively. In this paper, by virtue of classifying 4-cutwidth critical graphs reasonably, we shall attempt to characterize the inner structural features of the critical graphs with cutwidth-4 in detail.

Let $\mathcal{S}$${}_n=\{1,2,\cdots ,n\}$ for an integer $n>0$. The labeling of a graph $G=\left(V\right(G),E(G\left)\right)$ with $\left|V\right(G\left)\right|=n$ is a bijection $\pi :V\left(G\right)\to \mathcal{S}$${}_n$, viewed as an embedding of G into a path $P_n$ with vertices in $\mathcal{S}$${}_n$, where consecutive integers are the adjacent vertices. The cutwidth of G with respect to $\pi$ is

$$c(G,\pi )=\underset{1\le j<n}{max}\left|\{uv\in E\left(G\right):\pi \left(u\right)\le \pi <\pi \left(v\right)\}\right|,$$

(1)

which is also the congestion of the embedding. The cutwidth of G is defined to be

$$c\left(G\right)=\underset{\pi }{min}c(G,\pi ),$$

(2)

where the minimum is taken over all labelings $\pi$. If $k=c(G,\pi )$, then $\pi$, as well as the embedding induced by $\pi$, is called a k-cutwidth embedding of G. A labeling $\pi$ attaining the minimum in $\left(2\right)$ is an optimal labeling. For each i with $1\le i\le n$, let $u_i={\pi }^{-1}\left(i\right)$ and $S_j=\{u_1,u_2,\cdots ,u_j\}$. Define ${\nabla }_{\pi }\left(S_j\right)=\{u_i{u}_h\in E:i\le j<h\}$, which is called the (edge) cut at $[j,j+1]$ with respect to $\pi$. Using (2), we have

$$c(G,\pi )=\underset{1\le j<n}{max}\left|{\nabla }_{\pi }\left(S_j\right)\right|.$$

(3)

A $\pi$-max-cut of G is ${\nabla }_{\pi }\left(S_j\right)$, achieving the maximum in (3). For an optimal labeling $\pi$ of G with a $\pi$-max-cut ${\nabla }_{\pi }\left(S_{j_0}\right)$, if vertex $v_0={\pi }^{-1}\left(j_0\right)$ and $|{\nabla }_{\pi }\left(S_j\right)|\le k-2$ for every $1\le j\le j_0-1$ (or $j_0+1\le j<n$), then $v_0$ is called the small-cut vertex with respect to $\pi$.

For graph G and integer $i>0$, let $D_i\left(G\right)=\{v\in V\left(G\right):d_G\left(v\right)=i\}$ in which $d_G\left(v\right)$ is the degree of vertex $v\in V\left(G\right)$. Any vertex in $D_1\left(G\right)$ is called a pendant vertex in G. Any edge incident with a vertex in $D_1\left(G\right)$ is a pendant edge of G, and $E_p\left(G\right)=\{v_i{v}_j:v_i{v}_j\in E\left(G\right)and{v}_i{v}_{j}ispendant\}$ is a set of all pendant edges of G. For each $v\in V\left(G\right)$, let $N_G\left(v\right)=\{u\in V\left(G\right):uv\in E\left(G\right)\}$. If G possesses a vertex $v\in D_2\left(G\right)$ with $N_G\left(v\right)=\{v_1,v_2\}$ and $v_1{v}_2\notin E\left(G\right)$, then $G-v+v_1{v}_2$, the graph obtained from $G-v$ by adding a new edge $v_1{v}_2$, is called a series reduction of G. A graph H is a minor of G if H is obtained by deleting vertices, edges or carrying out series reductions in G and $c\left(H\right)=c\left(G\right)$. If $H,H^{\prime }$ are subgraphs of G, and $X\subseteq E\left(G\right)$, then, as in [1], $G\left[X\right]$ is an edge subgraph of G induced by X, $H\cup H^{\prime }=G[E\left(H\right)\cup E\left(H^{\prime }\right)]$ and $H\cup X=G\left[E\right(H)\cup X]$. Specifically, if $X=\left\{e\right\}$, then we write $G+e$ instead of $G\cup \left\{e\right\}$. Let G and $G^{\prime }$ be two disjoint graphs with $u\in V\left(G\right),v\in V\left(G^{\prime }\right)$; then, to identify u and v, denoted as $G{\oplus }_{u,v}{G}^{\prime }$, is to replace $u,v$ with a single vertex z (i.e.,$u=v=z)$ incident to all the edges which were incident to u and v, where z is called the identified vertex. Clearly, if $G^{\prime }=K_2$ with $K_2=u_0{u}_1$, then $G{\oplus }_{u,u_0}{K}_2=G{\oplus }_{u,u_0}{u}_0{u}_1=G+u_1+u_0{u}_1$. If graph G is 2-connected, then any two vertices of G lie on a common cycle. A subgraph decomposition $\mathcal{S}$ of G is a set of proper connected subgraphs $H_1,H_2,...,H_r$ of G whose union ${\bigcup }_{i=1}^r{H}_i$ is G, where $H_i,H_j\in \mathcal{S}$ are not necessarily edge-disjoint. A graph G is homeomorphically minimal if G does not have any series reductions. Two graphs G and H are homeomorphic if they can both be obtained from the same graph $\mathcal{G}$ by inserting new vertices of degree two into its edges. A graph G is said to be k-cutwidth critical if G is homeomorphically minimal with $c\left(G\right)=k$, such that every proper subgraph H of G satisfies $c\left(H\right)<k$. From definition, three properties of cutwidth below can be obtained immediately.

Lemma 1.

For graphs G and H, each of the following holds.

(1)

If H is a subgraph of G, then $c\left(H\right)\le c\left(G\right)$.

(2)

If H is homeomorphic to G, then $c\left(H\right)=c\left(G\right)$.

(3)

For a cut edge e in G, if $V_1,V_2$ are the vertex sets of two components of $G-e$, then there exists an optimal labeling $f^{*}$, such that the vertices in each of $V_1$ and $V_2$ are labeled consecutively.

Lemma 2

([8]). The unique 1-cutwidth critical graph is $K_2$. The only 2-cutwidth critical graphs are $K_3$ and $K_{1,3}$. All 3-cutwidth critical graphs are ${\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4$ and ${\tau }_5$ in Figure 1.

Figure 1. Five 3-cutwidth critical graphs.

Figure 1. Five 3-cutwidth critical graphs.

Lemma 3

([11]). For $k\ge 4$, a tree T is k-cutwidth critical if and only if T can be decomposed into three $(k-1)$-cutwidth subtrees, each of which is either a $(k-1)$-cutwidth critical tree or a sum of a $(k-1)$-cutwidth critical tree and a pendant edge.

Lemma 4

([12]). Let G be a k-cutwidth graph with a central vertex $v_0$ of $d_G\left(v_0\right)\ge 4$ and at least two cut edges $v_0{v}_1$ and $v_0{v}_2$. If G can be decomposed into three $(k-1)$-cutwidth graphs $G_1,G_2$ and $G_3$, then G is k-cutwidth critical if and only if each element of $\{G_i:1\le i\le 3\}$ is $(k-1)$-cutwidth critical.

The rest of this paper is organized as follows. Section 2 presents some preliminary results. Section 3 is focused on investigating 4-cutwidth critical graphs with a central vertex $v_0$. The characterizations of 4-cutwidth critical graphs with a central cycle $C_q$ ($q\ge 3$) are given in Section 4. Five 4-cutwidth critical graphs without a central vertex and a central cycle are discussed in Section 5. Furthermore, we give short concluding remarks in Section 6.

2. Preliminary Results

From [1], if $\mathcal{S}$ is a decomposition of a graph G, then $E\left(H_i\right)\cap E\left(H_j\right)=\varnothing$ for arbitrary $H_i,H_j\in \mathcal{S}$$(i\ne j)$, that is to say $H_i,H_j$ are edge-disjoint in G. In this article, for graph G and an integer $r>1$, if $G={\bigcup }_{i=1}^r{H}_i$ and there are at least two subgraphs $H_i,H_j$ such that $H_i,H_j$$(i\ne j)$ are edge-joint, then $\{H_i:1\le i\le r\}$ is also called a decomposition of G, also denoted by $\mathcal{S}$. For example, $\{{\tau }_4\left[\{x_1{x}_1^{\prime },x_1{x}_2,x_1{x}_3\}\right],{\tau }_4\left[\{x_2{x}_2^{\prime },x_2{x}_1,x_2{x}_3\}\right],$${\tau }_4\left[\{x_3{x}_3^{\prime },x_3{x}_2,x_3{x}_1\}\right]\}$ is an edge-joint decomposition of ${\tau }_4$, each of which is $K_{1,3}$ (see ${\tau }_4$ in Figure 1). Let $P_n=u_1{u}_2...u_n$ be a path with n vertices, such that for $1\le i<n$, $u_i$ and $u_{i+1}$ are adjacent vertices in $P_n$. By [9], $K_{1,2k-1}$ is k-cutwidth critical, so we let $d_G\left(v\right)\le 2k-2$ for each $v\in V\left(G\right)$. For G and $G^{\prime }$ which are homeomorphic, when no confusion occurs, if G is k-cutwidth critical after the series reductions are carried out, then we shall say that $G^{\prime }$ is also k-cutwidth critical. The following is immediate from Lemma 1:

$$\mathrm{if}v\in V\left(G\right),\mathrm{then}c(G-v)\le c\left(G\right).$$

(4)

Definition 1.

(i)

For graph G and integer $r>0$, let $v\in V\left(G\right)$ with $d_G\left(v\right)>r$. For $v_1,v_2,\dots ,v_r\in N_G\left(v\right)$, define $G(v;v_1,v_2,\dots ,v_r)$ to be the component of $G-\{v{v}_1,v{v}_2,\dots ,v{v}_r\}$ that contains v.

(ii)

Let $G_1,G_2$ be two disjoint graphs with $u\in V\left(G_1\right)$ and $v\in V\left(G_2\right)$. To identify u and v, denoted as $G_1{\oplus }_{u,v}{G}_2$, is to replace $u,v$ by a single vertex z$(i.e.,u=v=z)$ incident to all the edges which were incident to u and v, where z is called the identified vertex.

(iii)

Let $G_1$, $G_2$ and $G_3$ be three disjoint graphs, $D_3\left(K_{1,3}\right)=\left\{u_0\right\}$ and $D_1\left(K_{1,3}\right)=\{u_1,u_2,u_3\}$, $v_j\in V\left(G_j\right)$ for each $j\in \mathcal{S}$${}_3$. Define $K_{1,3}\circ (G_1,G_2,G_3)$ as the graph obtained from the disjoint union $G_1,G_2,G_3$ and $K_{1,3}$ by identifying $u_j$ with $v_j$ (again denoted as $v_j$) for each $j\in \mathcal{S}$${}_3$ (see Figure 3d in Section 3.1 below).

(iv)

Let $G_1$, $G_2$ and $G_3$ be three disjoint graphs, $P_3=u_1{u}_2{u}_3$ with $d_{P_3}\left(u_2\right)=2$ and $v_j\in V\left(G_j\right)$ for each $j\in \mathcal{S}$${}_3$. Define $P_3\circ (G_1,G_2,G_3)$ as the graph obtained from the disjoint union $G_1,G_2,G_3$ and $P_3$ by identifying $u_j$ with $v_j$ (again denoted as $v_j$) for each $j\in \mathcal{S}$${}_3$.

(v)

For $i\in \{1,2,\dots ,t\}$ with $t\ge 3$, let $G_i$ be a graph with $D_1\left(G_i\right)\ne \varnothing$ and $z_i\in D_1\left(G_i\right)$. Define $G={\oplus }_{z_0}(G_1,G_2,...,G_t)$ to be a graph obtained from disjoint union of $G_1,G_2,\dots ,G_t$ by identifying $z_1,z_2,\dots ,z_t$ into a single vertex $z_0$ in G. As $z_0=z_i$ in $G_i$, $z_0$ is viewed as the vertex $z_i$ in $G_i$.

(vi)

If $\left|V\right(G\left)\right|\ge 3$, then define $\mathcal{M}$$\left(G\right)=\{G-uv:uv\in E\left(G\right)anduvisnotacutedge\}\cup \{G-v:v\in D_1\left(G\right)\}$ to be the family of all proper maximal subgraphs of G.

Definition 2.

Suppose that vertex $v_0\in V\left(G\right)$ with $N_G\left(v_0\right)=\{v_1,v_2,\cdots ,v_p\}$, $v_0{v}_1,v_0{v}_2$ are two cut edges of G, $G_1^{\prime }=G(v_0;v_2,v_3,\cdots ,v_p)-v_0$, $G_2^{\prime }=G(v_0;v_1,v_2)$ and $G_3^{\prime }=G(v_0;v_1,v_3,\cdots ,v_p)-v_0$. For $i\in \mathcal{S}$${}_3$, let ${\pi }_i:V\left(G_i^{\prime }\right)\to \mathcal{S}$${}_{\left|V\right(G_i^{\prime }\left)\right|}$ be an optimal labeling of $G_i^{\prime }$, and let the labeling $\pi :V\left(G\right)\to \mathcal{S}$${}_n$ of G be as follows: for $v\in V\left(G\right)$,

$$\pi \left(v\right)=\left\{\begin{array}{ll}{\pi }_1\left(v\right)& ifv\in V\left(G_1^{\prime }\right),\\ {\pi }_2\left(v\right)+\left|V\left(G_1^{\prime }\right)\right|& ifv\in V\left(G_2^{\prime }\right),\\ {\pi }_3\left(v\right)+|V\left(G_1^{\prime }\right)|+|V\left(G_2^{\prime }\right)|& ifv\in V\left(G_3^{\prime }\right).\end{array}\right.$$

(5)

Then, the labeling π is called a labeling by the order $({\pi }_1,{\pi }_2,{\pi }_3)$ or $(V\left(G_1^{\prime }\right),V\left(G_2^{\prime }\right)$,$V\left(G_3^{\prime }\right))$.

Theorem 1

([12]). For any $v\in D_{\ge 3}\left(G\right)$, if there always are two vertices $v_1,v_2$ in $N_G\left(v\right)$ such that $v{v}_1,v{v}_2$ are cut edges in G, then $c\left(G\right)\le k$ if and only if $c\left(G(v;v_1,v_2)\right)\le k-1.$

Corollary 1.

For graph G, if there is a vertex $v\in D_{\ge 3}\left(G\right)$ such that $c\left(G(v;v_i,v_j)\right)\ge k-1$ holds for any $v_i,v_j\in N_G\left(v\right)$, then $c\left(G\right)\ge$k, where $v{v}_i,v{v}_j$ are both cut edges in G.

Lemma 5

([10]). Let graph G be k-cutwidth critical and $K_2=u_0{u}_1$. Then $c\left(G{\oplus }_{v_0,u_0}{K}_2\right)=k$ for $v_0\in V\left(G\right)$.

Theorem 2

([12]). With the notation of Definition 1$\left(iii\right)$, let at least one of $\{G_1,G_2,G_3\}$, say $G_2$, be $(k-1)$-cutwidth critical with $D_1\left(G_2\right)\ne \varnothing$. Then $c(K_{1,3}\circ (G_1,G_2,G_3))=k$.

Corollary 2

([12]). With the notation of Definition 1$\left(iii\right)$, for each $j\in \mathcal{S}$${}_3$, if $G_j$ is $(k-1)$-cutwidth critical with $v_j\in D_1\left(G_j\right)$, then $K_{1,3}\circ (G_1,G_2,G_3)$ is k-cutwidth critical.

Theorem 3.

With notation of Definition 1$\left(iv\right)$, if $c\left(G_j\right)=k-1$ for each $j\in \mathcal{S}$${}_3$, then $c(P_3\circ (G_1,G_2,G_3))=k$.

Proof. 

Let $G=P_3\circ (G_1,G_2,G_3)$. If $d_G\left(v_j\right)=2$ for $j=1$ or 3 then the series reductions are first carried out without effecting $c\left(G\right)=k$. As $G-\{v_2{v}_1,v_2{v}_3\}$ has three components $G_1,G_2$ and $G_3$ with cutwidth $k-1$, similar to that of (5), an optimal labeling $\pi :V\left(G\right)\to \mathcal{S}$${}_n$ obtained by the order $(V\left(G_1\right),V\left(G_2\right),V\left(G_3\right))$ satisfies $c(G,\pi )\le (k-1)+1=k$. Therefore, $c\left(G\right)\le k$ by (2). Additionally, it is not hard to verify that $c\left(G\right)\ge k$ by Corollary 1; this is because $c\left(G(v_2;v_i,v_j)\right)=k-1$ for any $v_i,v_j\in N_G\left(v_2\right)$. Hence $c\left(G\right)=k$, i.e., $c(P_3\circ (G_1,G_2,G_3))=k$. □

Corollary 3.

With notation of Definition 1 (iv), if the following hold:

(1)

$G_1,G_3$ are 2-connected;

(2)

$v_j$ is a small-cut vertex corresponding to an optimal labeling ${\pi }_j$ of $G_j$ for each $j\in \mathcal{S}$${}_3$;

(3)

$G_1,G_2$, $G_3$ are $(k-1)$-cutwidth critical, then $P_3\circ (G_1,G_2,G_3)$ is k-cutwidth critical, where $G_1,G_2,G_3$ are not necessarily distinct.

Proof. 

Let $G=P_3\circ (G_1,G_2,G_3)$. Since $N_G\left(v_2\right)=\{v_1,v_3\}$, $G(v_1;v_2)=G_1,G(v_2;v_1,v_3)=G_2$ and $G(v_3;v_2)=G_3$. First, $c\left(G\right)=k$ by Theorem 3. Second, we show $c\left(G^{\prime }\right)\le k-1$ for any $G^{\prime }\in$$\mathcal{M}$$\left(G\right)$, that is, G is k-cutwidth critical. Because any $G^{\prime }$ can be obtained by deleting a pendant edge $xy$ or an non-pendant edge $xy\in E\left(C_t\right)$ in G, $xy\notin \{v_2{v}_1,v_2{v}_3\}$, where $C_t$ is a cycle with length $t\ge 3$. There are two cases to consider: (1) $xy\in E\left(G_2\right)$; (2) $xy\in E\left(G_1\right)$ or $E\left(G_3\right)$. For Case (1), since $G_2$ is $(k-1)$-cutwidth critical, there is an optimal labeling ${\pi }_2^{\prime }$ such that $c(G_2-xy,{\pi }_2^{\prime })\le k-2$. Now, by Lemma 5, let ${\pi }_j^{\prime }$ be a labeling of $G_j$ such that $c\left(G_j{\oplus }_{u_j,v_j}{v}_j{v}_2\right)=k-1$ for $j=1,3$. Thus, a labeling $\pi$ of G by the order $({\pi }_1^{\prime },{\pi }_2^{\prime },{\pi }_3^{\prime })$ is obtained with $c(G-xy,\pi )\le k-1$ implying $c(G-xy)\le k-1$. For Case (2), let $xy\in E\left(G_3\right)$. By assumption, $c(G_3-xy)\le k-2$. Since $v_j$ is a small-cut vertex corresponding to an optimal labeling ${\pi }_j$ of $G_j$ for each $j\in \mathcal{S}$${}_3$, a labeling $\pi$ of G by the order $({\pi }_2,{\pi }_3,{\pi }_1)$ is obtained with $c(G-xy,\pi )\le k-1$ implying $c(G-xy)\le k-1$. Likewise, if $xy\in E\left(G_1\right)$ then $c(G-xy)\le k-1$ also. To sum up, G is k-cutwidth critical. □

Lemma 6.

Each graph in Figure 2 is 4-cutwidth critical.

Figure 2. Eight special 4-cutwidth critical graphs.

Figure 2. Eight special 4-cutwidth critical graphs.

Proof. 

Two steps can be used to finish the proof. For each $M_i$$(1\le i\le 8)$, Step 1 is used to show $c\left(M_i\right)=4$. This can be accomplished by two operations: $\left(1\right)$ $c(M_i,\pi )\ge 4$ for any labeling $\pi$ of $M_i$, which implies $c\left(M_i\right)\ge 4$; $\left(2\right)$ $M_i$ has an optimal labeling ${\pi }_0$ with $c(M_i,{\pi }_0)=4$. In Step 2, for any $M_i^{\prime }\in \mathcal{M}$$\left(M_i\right)$, $c\left(M_i^{\prime }\right)\le 3$ must be shown. Since operation of each of the two steps is easy, we omitted it here.

Let v be a cut-vertex with $d_G\left(v\right)\ge 3$ in G and $G_1,G_2,...,G_q$ be q connected components of $G-v$. Then, $G[V\left(G_i\right)\cup \left\{v\right\}]$$(1\le i\le q)$, denoted by $H_i$, is called the ith v-component of $G-v$. A vertex $v_0\in V\left(G\right)$ is called the central vertex of a k-cutwidth graph G if $v_0$ is a cut-vertex in G, such that all $v_0$-components of $G-v_0$ can form a decomposition $\mathcal{S}$ of G in which each element has equal cutwidth $\rho$ with $\rho <k$. For example, for graph ${\tau }_1$ in Figure 1, $H_i=K_2$$(1\le i\le 5)$ with edge $v_0{v}_i$ is the ith $v_0$-component of ${\tau }_1-v_0$; we can see that $\{H_1\cup H_2\cup H_3,H_1\cup H_2\cup H_4,H_1\cup H_2\cup H_5\}$ is a decomposition of ${\tau }_1$, each of which is a 2-cutwidth critical tree $K_{1,3}$, so $v_0$ is the central vertex of ${\tau }_1$. Likewise, each of $\{{\tau }_2,{\tau }_3\}$ has a decomposition of equal cutwidth-2 and a central vertex $v_0$ also, respectively.

For a cycle $C_q=x_1{x}_2...x_q{x}_1$ of G with $q\ge 3$ and $d_G\left(x_i\right)\ge 3$ for $1\le i\le q$, let $V\left(C_q\right)$ be a vertex-cut set of G. If $E\left(C_q\right)$ is also an edge-cut set of G and $G_i$ is the ith connected component of $G-E\left(C_q\right)$ leading from $x_i$, then $G[E\left(G_i\right)\cup \widetilde{E_i}]$, denoted by $H_i$, is called the ith $C_q$-component leading from $x_i$ of $G-E\left(C_q\right)$, where ${\tilde{E}}_i\subseteq E\left(C_q\right)$ and at least an ${\tilde{E}}_i\ne \varnothing$. A cycle $C_q$ with $q\ge 3$ is called a central cycle of a k-cutwidth graph G if $E\left(C_q\right)$ is an edge-cut set, such that one of the following is a decomposition $\mathcal{S}$ of G, each element of which has equal cutwidth $\rho$ with $\rho <k$,

(1)

$\{H_i:1\le i\le q\}$, or

(2)

$\{H_i^{\prime }:1\le i<q\}$ in which $H_i^{\prime }$ may be one of $\{H_i,H_{i-1}\cup H_i\cup H_{i+1}\}$ with $H_0=H_q,H_{q+1}=H_1$, and there exists at least $H_i^{\prime }\ne H_i$, or

(3)

$\{H_i^{″}:1\le i<q\}$, each of which is either $H_i$ or $H_{i-1}\left[E^{\prime }\right]\cup H_i\cup H_{i+1}\left[E^{″}\right]$ with $H_0=H_q,H_{q+1}=H_1$, and there exists at least $H_i^{″}\ne H_i$, where $H_{i-1}\left[E^{\prime }\right]\subset H_{i-1}$ and $H_{i+1}\left[E^{″}\right]\subset H_{i+1}$.

For example, in Figure 1, ${\tau }_4$ has a cycle $C_3=x_1{x}_2{x}_3{x}_1$, and ${\tau }_4-E\left(C_3\right)$ has three components $G_1,G_2,G_3$, each of which equals $K_2$. Let ${\tilde{E}}_1=\{x_1{x}_2,x_1{x}_3\},{\tilde{E}}_2=\{x_2{x}_1,x_2{x}_3\}$ and ${\tilde{E}}_3=\{x_3{x}_1,x_3{x}_2\}$, and let $H_i=G[E\left(G_i\right)\cup {\tilde{E}}_i]$ for $1\le i\le 3$. Then, $\{H_1,H_2,H_3\}$ is a decomposition $\mathcal{S}$ of ${\tau }_4$, in which each member is a 2-cutwidth critical subgraph $K_{1,3}$, and $C_3$ is the central cycle in ${\tau }_4$. For Case $\left(3\right)$, we take $M_5$ with a central cycle $C_3=x_1{x}_2{x}_3{x}_1$ in Figure 2 as an example. $M_5-E\left(C_3\right)$ also has three connected components $G_1,G_2,G_3$, which are $K_{1,3},K_{1,3},C_3$ and three $C_3$-components $H_1=G_1+x_1{x}_2+x_1{x}_3$ with $d_{G_1}\left(x_1^{\prime }\right)=d_{G_1}\left(x_1^{″}\right)=1$, $H_2=G_2+x_2{x}_1+x_2{x}_3$ with $d_{G_2}\left(x_2^{\prime }\right)=d_{G_2}\left(x_2^{″}\right)=1$ and $H_3=G_3+x_3{x}_1+x_3{x}_2$, respectively. Let $E^{\prime }=\{x_1{x}_1^{\prime },x_1{x}_1^{″}\},E^{″}=\{x_2{x}_2^{\prime },x_2{x}_2^{″}\}$, then $H_1\left[E^{\prime }\right]\subset H_1,H_2\left[E^{″}\right]\subset H_2$ and $\{H_1,H_2,H_1\left[E^{\prime }\right]\cup H_3\cup H_2\left[E^{″}\right]\}$ is a decomposition of equal cutwidth 3 of $M_5$, each member of which is also 3-cutwidth critical.

In the case that G is 2-connected and $E\left(C_q\right)$ is not an edge-cut set of G, suppose that $G-V\left(C_q\right)$ has q connected components $G_1,G_2,...,G_q$, with $V\left(G_i\right)\ne \varnothing$ for each $1\le i\le q$, and let $G[V\left(G_i\right)\cup \{x_i,x_{i+1}\}]$ be the ith 2-connected subgraph that contains edge $x_i{x}_{i+1}\in E\left(C_q\right)$. If $\{G[V\left(C_q\right)\cup V\left(G_i\right)]:1\le i\le q\}$ is a subgraph decomposition of equal cutwidth $\rho \le k-1$, then $C_q$ is also called the central cycle of G. For example, let $G=M_8$ with $C_3=x_1{x}_2{x}_3{x}_1$ in Figure 2. Clearly, $G-\{x_1,x_2,x_3\}$ has three components $y_1,y_2,y_3$, and $\{G\left[\{x_1,x_2,x_3,y_i\}\right]:1\le i\le 3\}=\{{\tau }_5,{\tau }_5,{\tau }_5\}$ is an edge-joint subgraph decomposition of equal cutwidth 3 of G. Hence, $C_3=x_1{x}_2{x}_3{x}_1$ is the central cycle of $M_8$. □

From Lemma 2, we have

Theorem 4.

For a 2-cutwidth critical graph $G\in \{K_{1,3},C_3\}$, one of the following holds:

(1)

G has a central vertex $v_0$, and $v_0$-components of $G-v_0$ constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each of which is $K_2$ with cutwidth 1;

(2)

G is a cycle $C_3$, whose three edges constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each element of which is $K_2$ with cutwidth 1.

Theorem 5.

For a 3-cutwidth critical graph $G\in \{{\tau }_i:1\le i\le 5\}$, one of the following holds:

(1)

has a central vertex $v_0$, and $v_0$-components of $G-v_0$ constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each of which equals $K_{1,3}$ or $C_3$ with cutwidth 2; or

(2)

G has a central cycle $C_3=x_1{x}_2{x}_3{x}_1$ with $d_G\left(x_i\right)$ = 3 for $x_i\in V\left(C_3\right)$, and $C_3$-components of $G-E\left(C_3\right)$ constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each member of which equals $K_{1,3}$ with cutwidth 2; or

(3)

G equals $C_4+x_1{x}_3$ or $C_4+x_2{x}_4$, where $C_4=x_1{x}_2{x}_3{x}_4{x}_1$ is a cycle of length 4.

3. 4-Cutwidth Critical Graphs with a Central Vertex

In this section, we shall verify the decomposability of the 4-cutwidth critical graphs with a central vertex. Since a k-cutwidth critical graph G is homeomorphically minimal, for the central cycle $C_q$$(q\ge 3)$ of G, we can let

$$d_G\left(v_i\right)\ge 3\mathrm{for\; every}{v}_i\in V\left(C_q\right).$$

(6)

3.1. 4-Cutwidth Critical Trees with a Central Vertex

Definition 3.

For a cut-vertex $v_0$ with $N_T\left(v_0\right)=\{v_i:1\le i\le qandq\ge 4\}$ in a tree T, let $H_i$ be a $v_0$-component of $T-v_0$ with $c\left(H_1\right)\ge c\left(H_2\right)\ge \dots \ge c\left(H_q\right)$ and $c\left({\bigcup }_{i=4}^q{H}_i\right)<k-1$, then define

$$T_i=\left\{\begin{array}{cc}{K}_{1,2k-3}\hfill & ifi<3and{H}_i=K_{1,2k-3},\hfill \\ H_i\cup \left(\bigcup _{i=4}^q{H}_i\right)\hfill & ifi<3and{H}_i\ne K_{1,2k-3},\hfill \\ H_3\cup \left(\bigcup _{i=4}^q{H}_i\right)\hfill & ifi=3.\hfill \end{array}\right.$$

(7)

If $c\left(T_i\right)=k-1$ for $1\le i\le 3$, then $\{T_1,T_2,T_3\}$ is called a subtree decomposition of equal cutwidth $k-1$ of T.

In Definition 3, for a decomposition $\{T_1,T_2,T_3\}$ of equal cutwidth $k-1$ of a k-cutwidth critical tree T, $E\left(T_{i_1}\right)\cap E\left(T_{i_2}\right)=E\left({\bigcup }_{i=4}^q{H}_i\right)$ ($1\le i_1\ne i_2\le 3$). If $E\left(T_{i_1}\right)\cap E\left(T_{i_2}\right)\ne \varnothing$, then $\{T_1,T_2,T_3\}$ is edge-joint; Otherwise $\{T_1,T_2,T_3\}$ is edge-disjoint.

There are eighteen 4-cutwidth critical trees in total by [9], each of which can be decomposed into three 3-cutwidth subtrees by Lemma 3. In fact, among these eighteen 4-cutwidth critical trees, each possesses one of the structures listed in Figure 3, in which $H_i\cup \left({\bigcup }_{i=4}^q{H}_i\right)$ is either one of ${\tau }_1$ and ${\tau }_2$ or homeomorphic to ${\tau }_2$ for $i=1,2,3$ in Figure 3a. $H_i\cup \left({\bigcup }_{i=4}^q{H}_i\right)$ is either ${\tau }_2$ or homeomorphic to ${\tau }_2$ for $i=2,3$ in Figure 3b, $H_i\cup \left({\bigcup }_{i=4}^q{H}_i\right)$ is either ${\tau }_2$ or homeomorphic to ${\tau }_2$ for $i=3$ in Figure 3c, either $H_i$ or $H_i-v_0{v}_i$ with $v_i\in N_{H_i}\left(v_0\right)$ is in $\{{\tau }_1,{\tau }_2\}$ for $i=1,2,3$ in Figure 3d. Thus, based on this, $M_1$ (see Figure 2) is 4-cutwidth critical, and again we have the following:

Theorem 6.

For a 4-cutwidth critical tree T, one of the following holds:

(1)

T possesses a configuration $K_{1,3}\circ (T_1,T_2,T_3)$ which can be decomposed into three edge-disjoint 3-cutwidth trees $T_1,T_2$ and $T_3$ (not necessarily distinct), and the 3-degree vertex of $K_{1,3}$ is the central vertex of T, where $T_i$ is a $v_0$-component of $T-v_0$ with either $T_i\in \{{\tau }_1,{\tau }_2\}$ or $T_i-v_0\in \{{\tau }_1,{\tau }_2\}$ for each $1\le i\le 3$ (see Figure 3d); or

(2)

T is a tree with a central vertex $v_0$ with $d_T\left(v_0\right)\ge 4$ and with an edge-joint decomposition $\{T_1,T_2,T_3\}$ of equal cutwidth 3, where $T_1,T_2$ and $T_3$ (not necessarily distinct), which are defined by (7), are either in $\{{\tau }_1,{\tau }_2\}$ or homeomorphic to ${\tau }_2$, and at least one of them, say $T_3$, is not ${\tau }_1$ (see Figure 3a–c, respectively).

Figure 3. Four structures of 4-cutwidth critical trees.

Figure 3. Four structures of 4-cutwidth critical trees.

3.2. 4-Cutwidth Critical Nontrees with a Central Vertex

We shall focus primarily on the structures of 4-cutwidth critical non-trees with a central vertex in this subsection.

Suppose now that $G_1,G_2$ and $G_3$ (not necessarily distinct) are mutually disjoint graphs, and at least one of them is not a tree. Let $K_{1,3}\circ (G_1^{\prime },G_2^{\prime },G_3^{\prime })$ be a graph obtained from the disjoint graphs $G_1^{\prime },G_2^{\prime },G_3^{\prime }$ and $K_{1,3}$ by identifying $u_i$ with $v_i$ (again denoted as $v_i$) for $i\in \mathcal{S}$${}_3$, where

$$G_i^{\prime }=\left\{\begin{array}{cc}{G}_i\hfill & if{v}_i\notin V\left(E_p\left(G_i\right)\right)\backslash D_1\left(G_i\right),\hfill \\ G_i-v_i^{\prime }\hfill & if{v}_i\in V\left(E_p\left(G_i\right)\right)\backslash D_1\left(G_i\right)and{v}_i{v}_i^{\prime }\in E\left(E_p\left(G_i\right)\right).\hfill \end{array}\right.$$

(8)

$u_i$ is a pendant vertex of $K_{1,3}$ and $v_i\in V\left(G_i\right)$ for $1\le i\le 3$. Obviously, if $c\left(G_1\right)=c\left(G_2\right)=c\left(G_3\right)$ and $D_3\left(K_{1,3}\right)=\left\{u_0\right\}$ then $u_0$ is the central vertex of $K_{1,3}\circ (G_1^{\prime },G_2^{\prime },G_3^{\prime })$.

Lemma 7.

Suppose that $G_i$ is $(k-1)$-cutwidth critical for $1\le i\le 3$, then $K_{1,3}\circ (G_1^{\prime },G_2^{\prime },G_3^{\prime })$ is a k-cutwidt critical graph, where $G_1,G_2,G_3$ are not necessarily distinct.

Proof. 

Let $G=K_{1,3}\circ (G_1^{\prime },G_2^{\prime },G_3^{\prime })$. If there exists at least a vertex $v_i\in N_G\left(u_0\right)$ such that $d_G\left(v_i\right)=2$, then the series reductions can be implemented first. Two cases need to be considered as follows.

Case 1. For $i\in \mathcal{S}$${}_3$, $v_i\notin V\left(E_p\left(G_i\right)\right)\backslash D_1\left(G_i\right)$.

By $\left(8\right)$, $G_i^{\prime }=G_i$ for $i\in \mathcal{S}$${}_3$. So $c\left(G_i^{\prime }\right)=c\left(G_i\right)=k-1$ by assumption, and $c(G_2^{\prime }+u_0{v}_2)=k-1$ by Lemma 5. Now, let ${\pi }_1,{\pi }_2,{\pi }_3$ be the labelings such that $c(G_1^{\prime },{\pi }_1)=k-1,c(G_2^{\prime }+u_0{v}_2,{\pi }_2)=k-1$ and $c(G_3^{\prime },{\pi }_3)=k-1$, respectively. Then, a labeling $\pi$ of G by the order $({\pi }_1,{\pi }_2,{\pi }_3)$ is obtained, and $c(G,\pi )=max\{c(G_1^{\prime },{\pi }_1),c(G_2^{\prime }+u_0{v}_2,{\pi }_2),c(G_3^{\prime },{\pi }_3)\}+1=(k-1)+1=k$, implying $c\left(G\right)\le k$. Since $u_0{v}_1,u_0{v}_2$ and $u_0{v}_3$ are cut-edges in G, $c(u_0;v_i,v_j)\ge k-1$ for any $v_i,v_j\in \{v_1,v_2,v_3\}$, leading to $c\left(G\right)\ge k$ by Corollary 1. Hence $c\left(G\right)=k$.

On the other hand, any $G^{\prime }\in \mathcal{M}$$\left(G\right)$ can be obtained by deleting a vertex y with degree one of a pendant edge $xy\notin E\left(C_t\right)$ or a nonpendant edge $xy\in E\left(C_t\right)$ in G, so $xy\ne u_0{v}_1,u_0{v}_2$ or $u_0{v}_3$, where $C_t=x_1{x}_2...x_t{x}_1$ is a cycle with length $t\ge 3$ in G. Without loss of generality, let $xy\in E\left(G_2^{\prime }\right)$. If $xy$ is pendant with $y\in D_1\left(G\right)$, then by the criticality of $G_2^{\prime }$, $c(G_2^{\prime }-y)\le k-2$ with a labeling ${\pi }_2^{\prime }$ such that $c(G_2^{\prime }-y,{\pi }_2^{\prime })\le k-2$. Since $G_1^{\prime }$ and $G_3^{\prime }$ are $(k-1)$-cutwidth critical, by $\left(6\right)$ in Lemma 5, two labelings ${\pi }_1^{\prime },{\pi }_3^{\prime }$ can be obtained such that $c(G_1^{\prime }{\oplus }_{u_1,v_1}{v}_1{u}_0,{\pi }_1^{\prime })=k-1$ with ${\pi }_1^{\prime }\left(u_0\right)=1$ and $c(G_3^{\prime }{\oplus }_{u_3,v_3}{v}_3{u}_0,{\pi }_1^{\prime })=k-1$ with $f_3^{\prime }\left(u_0\right)=\left|V\left(G_3^{\prime }\right)\right|+1$, respectively. Now, define $\pi :V\left(G^{\prime }\right)\to \{1,2,\dots ,|V\left(G^{\prime }\right)|-1\}$ to be a labeling of $G^{\prime }$ by the order $({\pi }_1^{\prime },{\pi }_2^{\prime },{\pi }_3^{\prime })$, then $c(G^{\prime },\pi )\le (k-2)+1=k-1$, i.e., $c\left(G^{\prime }\right)\le k-1$, meaning that G is k-cutwidth critical. Likewise, if $xy$ is not pendant with $xy\in E\left(C_t\right)$, then $c(G_2^{\prime }-xy)\le k-2$, and a labeling $\pi :V\left(G^{\prime }\right)\to \{1,2,\dots ,|V\left(G^{\prime }\right)\left|\right\}$ by the order $(G_1^{\prime }{\oplus }_{u_1,v_1}{v}_1{u}_0,G_2^{\prime }-xy,G_3^{\prime }{\oplus }_{u_3,v_3}{v}_3{u}_0)$ is also obtained, under which $c(G^{\prime },\pi )\le (k-2)+1=k-1$, i.e., $c\left(G^{\prime }\right)\le k-1$, meaning that G is also k-cutwidth critical. The cases of $xy\in E\left(G_1^{\prime }\right)$ or $E\left(G_3^{\prime }\right)$ are the same as that of $xy\in E\left(G_2^{\prime }\right)$, omitted here.

Case 2. There are at least a $v_{i_0}$, such that $v_{i_0}\in V\left(E_p\left(G_{i_0}\right)\right)\backslash D_1\left(G_{i_0}\right)$$(1\le i_0\le 3)$.

Three subcases need to be considered: $\left(1\right)$ there is unique $v_i$ (say $v_2$), such that $v_2\in V\left(E_p\left(G_2\right)\right)\backslash D_1\left(G_2\right)$; $\left(2\right)$ there are two $v_i^{\prime }s$ (say $v_1,v_3$), such that $v_1\in V\left(E_p\left(G_1\right)\right)\backslash D_1\left(G_1\right)$ and $v_3\in V\left(E_p\left(G_3\right)\right)\backslash D_1\left(G_3\right)$; $\left(3\right)$ $v_i\in V\left(E_p\left(G_i\right)\right)\backslash D_1\left(G_i\right)$ for each $1\le i\le 3$. For Subcase $\left(1\right)$, since $v_2\in V\left(E_p\left(G_2\right)\right)\backslash D_1\left(G_2\right)$, $G_2^{\prime }=G_2-v_2{v}_2^{\prime }$ with $v_2^{\prime }\in D_1\left(G_2\right)$. In this case, $G_2^{\prime }{\oplus }_{u_2,v_2}{u}_2{u}_0=G_2$, i.e., $G(u_0;v_1,v_3)=G_2$, $G_1^{\prime }=G_1$ and $G_3^{\prime }=G_3$ by $\left(9\right)$. Similarly, for Subcase $\left(2\right)$, $G_1^{\prime }{\oplus }_{u_1,v_1}{u}_1{u}_0=G_1$, $G_2^{\prime }=G_2$ and $G_3^{\prime }{\oplus }_{u_3,v_3}{u}_3{u}_0=G_3$; for Subcase $\left(3\right)$, $G_1^{\prime }{\oplus }_{u_1,v_1}{u}_1{u}_0=G_1$, $G_2^{\prime }{\oplus }_{u_2,v_2}{u}_2{u}_0=G_2$ and $G_3^{\prime }{\oplus }_{u_3,v_3}{u}_3{u}_0=G_3$. The remaining argument of any Subcase $\left(j\right)$ $(j=1,2,3)$ is similar to that of Case 1, omitted here. To sum up, G is k-cutwidth critical. □

Corollary 4.

Suppose that $G_i\in \{{\tau }_i:1\le i\le 5\}$ for $1\le i\le 3$, then $K_{1,3}\circ (G_1^{\prime },G_2^{\prime },G_3^{\prime })$ is a 4-cutwidt critical graph, where at least a $G_i\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$, and $G_1,G_2,G_3$ are not necessarily distinct.

Corollary 5.

Suppose that $G_i\in \{{\tau }_i:1\le i\le 4\}$ for $1\le i\le 3$, then ${\oplus }_{u_0}(G_1,G_2,G_3)$ is a 4-cutwidt critical graph, where at least a $G_i\in \{{\tau }_3,{\tau }_4\}$, and $G_1,G_2,G_3$ are not necessarily distinct.

Lemma 8.

Let $P_3=u_1{u}_2{u}_3$, $G_i$ be 3-cutwidth critical with $v_i\in V\left(G_i\right)$ for $1\le i\le 3$ and satisfy the following:

(i)

each non cut-edge of $G_2$ may be subdivided once, and $v_2$ may possibly be the subdivision vertex;

(ii)

$G_2\ne {\tau }_1$;

(iii)

if $G_2\in \{{\tau }_2,{\tau }_3\}$, then $v_2$ is not either the central vertex or the pendant vertex of it;

(iv)

$G_i\notin \{{\tau }_2,{\tau }_3\}$ for $i=1$ or 3 if $G_2\in \{{\tau }_2,{\tau }_3\}$.

Then, $P_3\circ (G_1^{\prime },G_2,G_3^{\prime })$ is a 4-cutwidt critical graph, where $G_1,G_2,G_3$ are not necessarily distinct, and at least one of them is not in $\{{\tau }_1,{\tau }_2\}$.

Proof. 

Let $G=P_3\circ (G_1^{\prime },G_2,G_3^{\prime })$. By assumption, for $i=1,3$, $G_i^{\prime }=G_i$ with $v_i\notin V\left(E_p\left(G_i\right)\right)\backslash D_1\left(G_i\right)$ or $G_i-v_i^{\prime }$ with $v_i{v}_i^{\prime }\in E\left(E_p\left(G_i\right)\right)$ and $v_i^{\prime }\in D_1\left(G_i\right)$. So, $H_i=G_i$ or $G_i+v_i{v}_2$ for $i=1,3$ and $H_i=G_2$ for $i=2$. Thus, with an argument similar to that of Lemma 7, G is 4-cutwidth critical. □

Suppose that $G_1\in \{{\tau }_2,{\tau }_3\}$ with the central vertex $u_1$(=$v_0$) and two cut edge $u_1{v}_1,u_1{v}_2$, such that any $u_1$-component of $G_1-u_1$ is 2-cutwidth critical (see ${\tau }_1-{\tau }_5$ in Figure 1). For any 3-cutwidth nontree graph $G_2\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$ with cycle $C_3=x_1{x}_2{x}_3{x}_1$, if there is a vertex (say $x_1$) in $C_3$ such that $\left(1\right)$ $x_1\ne u_2$ when $G_2={\tau }_3$ with the central vertex $u_2$(=$v_0$); or $\left(2\right)$ if $F_1$ is a component of $G_2-E\left(C_3\right)$ leading from $x_1$, then either $F_1=x_1{x}_1^{\prime }$ with $d_{G_2}\left(x_1^{\prime }\right)=1$ when $G_2={\tau }_4$ or $F_1=x_1$ only when $G_2={\tau }_5$; or $\left(3\right)$ $G_1$ and $G_2$ are not necessarily distinct; or $\left(4\right)$ if $G_1={\tau }_3$ and $G_2={\tau }_5$, then $d_{G_2}\left(x_1\right)=2$. Only then, by $\left(8\right)$, do we have

Lemma 9.

Graph $G_1{\oplus }_{u_1,x_1}{G}_2^{\prime }$ is a 4-cutwidth critical graph.

Proof. 

Let $G=G_1{\oplus }_{u_1,x_1}{G}_2^{\prime }$ with optimal labeling $\pi$, and ${\pi }_1$ be a sublabeling of $\pi$ restricted on $G_1$. By assumption, $G_1-u_1$ has three $u_1$-components $H_1,H_2$ and $H_3$, each of which is either $K_{1,3}$ or $C_3$ by Theorem 3. Suppose that ${\pi }_1$ is obtained by the order $({\pi }_1^{\prime },{\pi }_2^{\prime },{\pi }_3^{\prime })$ with $max\{{\pi }_1^{\prime }\left(v\right):v\in V(H_1-u_1)\}<{\pi }_2^{\prime }\left(v\right)<min\{{\pi }_3^{\prime }\left(v\right):v\in V(H_3-u_1)\}$ for $v\in V\left(H_2\right)$ if ${\pi }_1^{\prime },{\pi }_2^{\prime },{\pi }_3^{\prime }$ are optimal labelings of $H_1-u_1,H_2$ and $H_3-u_1$, respectively. Without loss of generality, let $H_1=K_{1,3}$ with cutwidth 2. Since $G_2$ is 3-cuwidth critical and $x_1\in V\left(C_3\right)$ in $G_2$, whether $G_2={\tau }_4$ or $G_2={\tau }_i$ with $i=3,5$, if ${\nabla }_{\pi }\left(S_j\right)$ is a $\pi$-max-cut of G, then $j<\pi \left(u_1\right)$ and $|{\nabla }_{\pi }\left(S_j\right)|=4$. Hence, $c\left(G\right)\le 4$. On the other hand, assuming that $u_1{v}_1,u_1{v}_2$ are cut edges in G, $\pi (u_1;v_1,v_2)=K_{1,3}{\oplus }_{u_1,x_1}{G}_2^{\prime }$ when $G_1={\tau }_2$ or $C_3{\oplus }_{u_1,x_1}{G}_2^{\prime }$ when $G_1={\tau }_3$, so $c\left(\pi (u_1;v_1,v_2)\right)\ge 3$, resulting in $c\left(G\right)\ge 4$ by Corollary 1. Thus, $c\left(G\right)=4$.

We now verify that G is 4-cutwidth critical. For any edge $e\in E\left(G\right)$, e is in either $E\left(G_1\right)$ or $E\left(G_2\right)$. Since $G_1\in \{{\tau }_2,{\tau }_3\}$ which is 3-cutwidth critical, if $e\in E\left(G_1\right)$ then we can always find a labeling ${\overline{\pi }}_1$ of $G_1-e$ such that $c(G_1-e)=3$ and ${\overline{\pi }}_1\left(u_1\right)=\left|V\left(G_1\right)\right|$. For $G_2\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$, we can always find an optimal labeling ${\overline{\pi }}_2$ of $G_2^{\prime }$ such that ${\overline{\pi }}_2\left(x_1\right)=1$. Thus, a labeling $\overline{\pi }$ of $G-e$ by the order $({\overline{\pi }}_1,{\overline{\pi }}_2)$ is obtained with $c(G-e,\overline{\pi })=3$ leading to $c(G-e)\le 3$. Similarly, if $e\in E\left(G_2\right)$ then $c(G-e)\le 3$ also. This completes the proof. □

From Lemma 9, we can see that if a critical non-tree G with cutwidth 4 can be decomposed into two 3-cutwidth critical subgraphs $G_1\in \{{\tau }_2,{\tau }_3\}$ and $G_2\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$, then $\left(1\right)$G has a central vertex $u_1$; $\left(2\right)$ $u_1$ is also the central vertex of at least one of $G_1$ and $G_2$. For example, let $K_{1,3}^{\left(j\right)}$ with a pendant vertex $y^{\left(j\right)}$$(1\le j\le 4)$ be the copy of $K_{1,3}$ with a pendant vertex y, and $y^{\left(j\right)}$ be the copy of y, $y^{\left(0\right)}$ be a vertex of a 3-cycle $C_3^{\prime }$. Then, graph ${\oplus }_{y_0}(K_{1,3}^{\left(1\right)},...,K_{1,3}^{\left(4\right)},C_3^{\prime })$, obtained by identifying $y^{\left(0\right)},y^{\left(1\right)},...,y^{\left(4\right)}$ into a vertex $y_0$$(i.e.,y_0=y^{\left(0\right)}=...=y^{\left(4\right)})$, is a 4-cutwidth critical graph with the central vertex $y_0$, and this graph can be decomposed into two 3-cutwidth critical subgraphs ${\oplus }_{y_0}(K_{1,3}^{\left(1\right)},K_{1,3}^{\left(2\right)},K_{1,3}^{\left(3\right)})$ $(={\tau }_2)$ with the central vertex $y_0$, and ${\oplus }_{y_0}(K_{1,3}^{\left(3\right)},K_{1,3}^{\left(4\right)},C_3^{\prime })$ $(={\tau }_3)$ with the central vertex $y_0$; $\left(3\right)$ there are at least two cut edges $u_1{v}_1,u_1{v}_2$.

Lemma 10.

Let G be a 4-cutwidth critical nontree graph with the central vertex $v_0$ and at least two cut edges $v_0{v}_1$ and $v_0{v}_2$. If G can be decomposed into two 3-cutwidth graphs $G_1,G_2$ (not necessarily distinct), then the following hold:

(1)

$G_1,G_2$ are in $\{{\tau }_i:2\le i\le 5\}$;

(2)

at least one of $G_1$ and $G_2$, say $G_1$, is in $\{{\tau }_2,{\tau }_3\}$, while $G_2\ne {\tau }_2$;

(3)

$v_0$ is the central vertex of $G_1$, but $v_0$ is only a vertex of any 3-cycle $C_3$ of $G_2$.

Proof. 

Since G is a non-tree graph, we do not consider the cases that $G_1$ and $G_2$ are both ${\tau }_1$ or ${\tau }_2$. We first show that $G_1,G_2$ are in $\{{\tau }_i:2\le i\le 5\}$ by contradiction. Suppose that there is some $G_i$ (say $G_2$) such that $c\left(G_2\right)=3$ but $G_2$ is not 3-cutwidth critical, then there is at least a pendant edge $xy\in E\left(G_2\right)$ with $y\in D_1\left(G\right)$ or a non-pendant edge $xy\in E\left(C_t\right)$ such that $c(G_2-y)=3$ or $c(G_2-xy)=3$, respectively, where $C_t=x_1{x}_2...x_t{x}_1$ is a cycle with length $t\ge 3$. For the former, because $c\left(G_1\right)=3$ by assumption, $c(G-y)=4$ by Lemma 9. Likewise, for the latter, $c(G-xy)=4$ also by Lemma 9. All are contrary to the criticality of G. Hence $G_1,G_2$ are both in $\{{\tau }_i:2\le i\le 5\}$.

Next, by the assumption that $v_0$ is the central vertex and $v_0{v}_1$ and $v_0{v}_2$ are both cut edges in G, we claim that at least one of $G_1$ and $G_2$ (say $G_1$) must be ${\tau }_2$ or ${\tau }_3$. This is because otherwise, there is at most a vertex $v_1\in N_G\left(v_0\right)$ such that $v_0{v}_1$ is a cut edge in G if $G_1$ and $G_2$ are both in $\{{\tau }_4,{\tau }_5\}$, which is a contradiction. So $\left(2\right)$ holds and $G_1\in \{{\tau }_2,{\tau }_3\}$.

Third, assume that $v_0$ is neither the central vertex of $G_1$ nor a vertex of a 3-cycle $C_3$ of $G_2$ if $G_2={\tau }_3$. Without loss of generality, let $G_1={\tau }_2$. Then $G_2$ is either ${\tau }_3$ or one of $\{{\tau }_4,{\tau }_5\}$. For $G_2={\tau }_3$, by assumption, $v_0$ is not also the central vertex of $G_2$. Thus, except three vertices of 3-cycle $C_3$ of $G_2$, three cases need to be considered: $\left(a\right)$ $v_0$ is not only a subdivision vertex of some non-pendant edge in $G_1$ but also a subdivision vertex of some non-pendant cut edge in $G_2$; $\left(b\right)$ $v_0$ is a subdivision vertex of some non-pendant edge in $G_1$, but $v_0$ is a nonpendent vertex of $G_2$; $\left(c\right)$ $v_0$ is not only a non-pendant vertex of $G_1$ but also a non-pendant vertex of $G_2$. For any case of Cases $\left(a\right)$–$\left(c\right)$, we can easily verify that $c\left(G\right)=3$ by Lemma 1(3) and Theorem 5, contrary to the assumption of $c\left(G\right)=4$. Likewise, for $G_2\in \{{\tau }_4,{\tau }_5\}$, there are only two cases to consider: ${\left(a\right)}^{\prime }$ $v_0$ is a subdivision vertex of some non-pendant edge in $G_1$, but $v_0$ is a arbitrary vertex of $G_2$; ${\left(b\right)}^{\prime }$ $v_0$ is a nonpendant vertex of $G_1$, but $v_0$ is a arbitrary vertex of $G_2$. Furthermore, in any case, $c\left(G\right)=3$, also a contradiction. This completes the proof. □

For a cut-vertex $v_0\in D_{\ge 4}\left(G\right)$ graph G and all $v_0$-components $H_i=G[V\left(G_i\right)\cup \left\{v_0\right\}]$$(1\le i\le m)$ of $G-v_0$, we define a decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ each of which has cutwidth 3 below. Let $E_0$ be an edge subset taken from ${\overline{G}}_3$ such that the cutwidth of the connected subgraph $H_i\cup G\left[E_0\right]$ is 3 if $c\left(H_i\right)<3$, for $1\le i<3$. Then, we obtain the following:

Definition 4.

For a cut-vertex $v_0\in D_{\ge 4}\left(G\right)$ of G and the $v_0$-component $H_i$$(1\le i\le q)$ of $G-v_0$, $min\{c\left(H_i\right):1\le i\le 3\}\ge max\{c\left(H_i\right):4\le i\le q\}$ and the cutwidth of ${\bigcup }_{i=3}^q{H}_i$ is three. For $1\le i\le 3$, define

$${\overline{G}}_i=\left\{\begin{array}{cc}{H}_i\hfill & ifi<3andc\left(H_i\right)=3,\hfill \\ H_i\cup G\left[E_0\right]\hfill & ifi<3andc\left(H_i\right)<3,\hfill \\ \bigcup _{i=3}^q{H}_i\hfill & ifi=3.\hfill \end{array}\right.$$

(9)

If $c\left({\overline{G}}_i\right)=3$ for $i=1,2$, then $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ is called a decomposition of equal cutwidth 3 of G, and G is called a graph with a central vertex $v_0$, where $E_0$ is an edge subset taken from ${\overline{G}}_3$ such that $c(H_i\cup G\left[E_0\right])=3$ if $c\left(H_i\right)<3$ for $i=1,2$

Lemma 11.

Let G be a 4-cutwidth critical graph with the central vertex $v_0$ and at least two cut edges $v_0{v}_1$ and $v_0{v}_2$. If G can be decomposed into three 3-cutwidth graphs ${\overline{G}}_1,{\overline{G}}_2$ and ${\overline{G}}_3$, then G is 4-cutwidth critical if and only if each of $\{{\overline{G}}_i:1\le i\le 3\}$ is either a 3-cutwidth critical graph or homeomorphic to a 3-cutwidth critical nontree graph, and $v_0$ is not the central vertex of ${\overline{G}}_i$ if ${\overline{G}}_i\in \{{\tau }_1,{\tau }_2,{\tau }_3\}$.

Proof. 

The proof is straightforward using Lemma 4, omitted here. □

Lemma 12.

For a 4-cutwidth graph G with a central vertex $v_0\in V\left(G\right)$, if $G-v_0$ has at least three $v_0$-component $H_i^{\prime }s$ and each $H_i$ is 2-connected in G, then G is 4-cutwidth critical if and only if $G=M_2$ (see $M_2$ in Figure 2).

Proof. 

Sufficiency: this is obvious using Lemma 6.

Necessity: By assumption, for any vertex $v_i\in N_G\left(v_0\right)$, $v_i\in V\left(C_{t_i}\right)$, where $C_{t_i}$$(t_i\ge 3)$ is a cycle of $H_i$ and $V\left(C_{t_i}\right)\cap V\left(C_{t_j}\right)=\left\{v_0\right\}$ for any $i\ne j$ only. Since $C_3$ is a minor of any $C_{t_i}$ and $c\left(G\right)=4$, $M_2$ with cutwidth 4 is a minor. Hence $G=M_2$ by the criticality of G. □

Lemma 13.

For a 4-cutwidth critical non-tree graph G with a central vertex $v_0\in V\left(G\right)$, G has a subgraph decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$, in which ${\overline{G}}_i$ is 2-cutwidth critical for $i=1$, one of whose pendant vertices is $v_0$, and 3-cutwidth critical for $i=2,3$ if and only if G is one of graphs $M_9$–$M_{17}$ in Figure 4, where ${\overline{G}}_1=K_{1,3},{\overline{G}}_i\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$ for $i=2,3$ with ${\tau }_4=H_2+v_{0}x$ (see Figure 4).

Proof. 

Similar to that of Lemma 6, we can show that graphs $M_9$–$M_{17}$ in Figure 4 are all 4-cutwidth critical.

Sufficiency: For graph $M_9$, let ${\overline{G}}_1=K_{1,3}$, ${\overline{G}}_2=H_2+v_{0}x={\tau }_4$ and ${\overline{G}}_3=H_3+v_{0}x={\tau }_4$, $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ is the subgraph decomposition desired. Likewise, for graphs $M_{10}$–$M_{12}$, let ${\overline{G}}_1=K_{1,3}$, ${\overline{G}}_2=H_2+v_{0}x={\tau }_4$, ${\overline{G}}_3=H_3\in \{{\tau }_3,{\tau }_5\}$ with $d_{{\tau }_3}\left(v_0\right)=2$ and $d_{{\tau }_5}\left(v_0\right)=2$ or 3, respectively; for graphs $M_{13}$–$M_{17}$, let ${\overline{G}}_1=K_{1,3}$, ${\overline{G}}_2=H_2\in \{{\tau }_3,{\tau }_5\}$ with $d_{{\tau }_3}\left(v_0\right)=d_{{\tau }_5}\left(v_0\right)=2$, ${\overline{G}}_3=H_3\in \{{\tau }_3,{\tau }_5\}$ with $d_{{\tau }_3}\left(v_0\right)=2$ and $d_{{\tau }_5}\left(v_0\right)=2$ or 3, respectively, $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ is the subgraph decomposition desired.

Necessity: Suppose by contradiction that $G\notin \{M_i:9\le i\le 17\}$. By assumption, ${\overline{G}}_i\in \{K_{1,3},C_3\}$ for $i=1$, and $\{{\tau }_i:1\le i\le 5\}$ for $i=2,3$. Three cases, which are at least a ${\overline{G}}_i={\tau }_1$ for $i=2,3$, ${\overline{G}}_2={\overline{G}}_3={\tau }_2$ and ${\overline{G}}_2={\overline{G}}_3={\tau }_5$ with $d_{{\overline{G}}_2}\left(v_0\right)=d_{{\overline{G}}_3}\left(v_0\right)=3$, respectively, can be first excluded; this is because that G either is a tree or is not 4-cutwidth critical in these cases, which is a contradiction. Thus, noting that 3-cutwidth critical subgraphs ${\overline{G}}_2$, ${\overline{G}}_3$ are symmetrical in G and $c\left(G\right)=4$ is sufficient to verify two cases:

(1)

${\overline{G}}_1=K_{1,3}$, one of whose three pendant vertices is $v_0$, ${\overline{G}}_2\in \{{\tau }_2,{\tau }_3\}$ and ${\overline{G}}_3\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$;

(2)

${\overline{G}}_1=C_3$, one of whose three 2-degree vertices is $v_0$, ${\overline{G}}_2\in \{{\tau }_2,{\tau }_3\}$ and ${\overline{G}}_3\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$.

By assumption, we do not consider the following five subcases contained in cases (1) and (2), respectively:

(a1)

$d_G\left(v_0\right)\ge 7$ because of $c\left(K_{1,7}\right)=4$;

(a2)

$M_2$ is a subgraph of G because of $c\left(M_2\right)=4$;

(a3)

G is a tree because G is a non-tree graph;

(a4)

$\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ is a decomposition of equal cutwidth 3;

(a5)

$c\left(G\right)=3$ because G is 4-cutwidth critical.

Based on this, for cases (1) and (2), we only consider vertices $u_0$, x of ${\tau }_2$, vertices $u_0,x,y$ of ${\tau }_3$, vertex $x_1$ of ${\tau }_4$ and vertices $x_1,x_2$ of ${\tau }_5$ (see Figure 1), respectively, which may be the central $v_0$ of G. For convenience, let ${\overline{G}}_2\in \{{\tau }_2^{u_0},{\tau }_2^x,{\tau }_3^{u_0},{\tau }_3^x,{\tau }_3^y\}$, ${\overline{G}}_3\in \{{\tau }_3^{u_0},{\tau }_3^x,{\tau }_3^y,{\tau }_4^{x_1},{\tau }_5^{x_1},{\tau }_5^{x_2}\}$, where ${\tau }_2^{u_0},{\tau }_2^x$ are copies of ${\tau }_2$ corresponding to $u_0,x$ of ${\tau }_2$, ${\tau }_3^{u_0},{\tau }_3^x,{\tau }_3^y$ are copies of ${\tau }_3$ corresponding to $u_0,x,y$ of ${\tau }_3$; ${\tau }_4^{x_1}$ is a copy of ${\tau }_4$ corresponding to $x_1$ of ${\tau }_4$, and ${\tau }_5^{x_1}$ and ${\tau }_5^{x_2}$ are copies of ${\tau }_5$ corresponding to $x_1,x_2$ of ${\tau }_5$, respectively. In this case, we can see that there are at least a $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$, which is a decomposition of one of $\{M_i:9\le i\le 17\}$ not considered here. For example, $\{K_{1,3},{\tau }_3^y,{\tau }_3^y\}$ is a decomposition of $M_{15}$. So, by at most $2\times C_5^1\times C_6^1-1$ direct operations and at most $2\times C_5^1\times C_6^1-1$ computations without considering $G\in \{M_i:9\le i\le 17\}$ by assumption, we can see that G is not 4-cutwidth critical, which is a contradiction. Hence, $G\in \{M_i:9\le i\le 17\}$. □

From Lemmas 7–13, we have:

Theorem 7.

For a 4-cutwidth non-tree graph G with a central vertex $v_0$, G is 4-cutwidth critical if and only if G has one of the following six configurations.

(1)

For $1\le i\le 3$, if $G_i$ is some ${\tau }_i$$(1\le i\le 5)$ in Figure 1 and $G_i^{\prime }$ corresponding to $G_i$ is a graph defined in $\left(8\right)$, then $G=K_{1,3}\circ (G_1^{\prime },G_2^{\prime },G_3^{\prime })$, where $G_1,G_2$ and $G_3$ are not necessarily different;

(2)

$G=P_3\circ (G_1^{\prime },G_2,G_3^{\prime })$, where $G_i\in \{{\tau }_i:1\le i\le 5\}$ with $v_i\in V\left(G_i\right)$ for $1\le i\le 3$ and $G_i^{\prime }$ corresponding to $G_i$ is a graph defined in $\left(8\right)$, $G_i\notin \{{\tau }_2,{\tau }_3\}$ for $i=1,3$ and $G_i\ne {\tau }_1$ for $i=2$ $v_2$ is not either the central vertex or the pendent vertex when $G_2\in \{{\tau }_2,{\tau }_3\}$ but $v_2$ is possible to a subdivision vertex of a non cut-edge of $G_2$ when $G_2\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$;

(3)

$G=G_1{\oplus }_{u_1,x_1}{G}_2^{\prime }$ with the central vertex $u_1$ of $d_G\left(u_1\right)<7$, where $G_1\in \{{\tau }_2,{\tau }_3\}$ with the central vertex $u_1$$(u_1=v_0$ of ${\tau }_2$ or ${\tau }_3$, respectively, see Figure 1$),$ $G_2\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$ with a 3-cycle $C_3\subset G_2$ and $x_1\in V\left(C_3\right)$ with $d_{G_1}\left(u_1\right)+d_{G_2}\left(x_1\right)\le 6$, $G_2^{\prime }$ corresponding to $G_2$ is a graph defined in $\left(8\right)$;

(4)

G has a subgraph decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ of equal cutwidth 3, defined in Definition 4, where G is a graph with a central vertex $v_0$ of $d_G\left(v_0\right)\ge 4$ and at least two cut edges $v_0{v}_1,v_0{v}_2$, ${\overline{G}}_i$ is 3-cutwidth critical for $1\le i\le 3$;

(5)

G has a subgraph decomposition $\{C_3,C_3^{\prime },C_3^{″}\}$ of equal cutwidth 2, each of which is a $v_0$-component of $G-v_0$, where $v_0$ is the central vertex $v_0$ of degree 6 of G, and $C_3^{\prime }$ and $C_3^{″}$ are the copies of a 3-cycle $C_3$;

(6)

G is one member of $\{M_i:9\le i\le 17\}$ with a central vertex $v_0$ (see Figure 4) and a subgraph decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$, in which ${\overline{G}}_1=K_{1,3}$, one of whose pendant vertices is $v_0$, ${\overline{G}}_i\in \{{\tau }_3,{\tau }_4,{\tau }_5\}$ for $i=2,3$, where ${\overline{G}}_i$ satisfies:

(i)

$v_0$ is a 2-degree vertex y of $C_3$ of ${\overline{G}}_i$ for ${\overline{G}}_i={\tau }_3$;

(ii)

if the 3-degree vertex of ${\overline{G}}_1(=K_{1,3})$ is x and ${\overline{G}}_i={\tau }_4$, then ${\tau }_4=H_2+v_{0}x$ and $v_0$ is a 3-degree vertex of ${\overline{G}}_i$;

(iii)

$v_0$ is either a 2-degree vertex of ${\overline{G}}_i$ or a 3-degree vertex of ${\overline{G}}_i$ for ${\overline{G}}_i={\tau }_5$, but if ${\overline{G}}_2={\overline{G}}_3={\tau }_5$ and $v_0$ is a 3-degree vertex of ${\overline{G}}_2$, then $v_0$ must not be a 3-degree vertex of ${\overline{G}}_3$, and vice versa.

4. 4-Cutwidth Critical Graphs with a Central Cycle

In this section, we aim to investigate 4-cutwidth critical graphs with a central cycle $C_q=x_1{x}_2...x_q{x}_1$ with $q\ge 3$.

Lemma 14.

Assume that graph G is 4-cutwidth critical with a central cycle $C_q$ of length q, then $q\le 6$.

Proof. 

Assume, contrary to that, that $q\ge 7$ and $G_i$ is the ith connected component leading from $x_i$ of $G-E\left(C_q\right)$. Without loss of generality, let $q=7$, i.e., $C_7=x_1{x}_2...x_7{x}_1$ with $d_G\left(x_i\right)\ge 3$ for $1\le i\le 7$ (see an example in Figure 5a), and let $\pi :V\left(G\right)\to \mathcal{S}$${}_n$ be an optimal 4-cutwidth labeling with $min\{\pi \left(v\right):v\in V\left(G_1\right)\}=\pi \left(x_1\right)<\pi \left(x_7\right)<\pi \left(x_2\right)<\pi \left(x_6\right)<\pi \left(x_3\right)<\pi \left(x_5\right)<\pi \left(x_4\right)=max\{\pi \left(v\right):v\in V\left(G_4\right)\}$. By the criticality of G, we may always assume that $G_i\in \{K_2,K_{1,3},C_3\}$. By direct computations, there are at least three $G_i^{\prime }$s (say $G_1,G_4$ and $G_6$) such that $G_1\ne K_2,G_4\ne K_2$ and $G_6\ne K_2$. Otherwise, $c\left(G\right)=3$, contrary to $c\left(G\right)=4$. Since G is 4-cutwidth critical, we can let $G_1,G_4,G_6\in \{K_{1,3},C_3\}$, say $G_1=C_3$ and $G_4=G_6=K_{1,3}$ (see Figure 5a). In this case, $c\left(G\right)=4$ and $c(G-x_i^{\prime })=4$ for any $G_i=K_2=x_i{x}_i^{\prime }$ with $i=2,3,5$, contrary to the criticality of G. On the other hand, there is at least a 4-cutwidth critical graph G with a central cycle $C_6=x_1{x}_2...x_6{x}_1$ such that $G_1=G_3=G_5=K_{1,3}$, $G_2=G_4=G_6=K_2$ and $d_G\left(x_i\right)=3$ for $1\le i\le 6$ (see Figure 5b). Hence $q\le 6$. □

From Lemma 14, in the sequel, we shall characterize the 4-cutwidth critical graphs with a central cycle of lengths 3–6, respectively.

4.1. Graphs with a Central Cycle of Length Three

Definition 5.

Let $C_3=x_1{x}_2{x}_3{x}_1$ be the central cycle of G, $G_i$$(1\le i\le 3)$ be the ith connected component leading from $x_i$ of $G-E\left(C_3\right)$, and $x_1,x_2,x_3$ be cut vertices in G. Then, for $1\le i\le 3$, define

$$H_i=\left\{\begin{array}{cc}{G}_i\hfill & ifc\left(G_i\right)=3butc((G_i-xy)\cup G\left[E^{\prime }\right])<3for{E}^{\prime }\subseteq E\left(C_3\right)withxy\in E\left(G_i\right),\hfill \\ G_i\hfill & ifc(G_i\cup G\left[E^{″}\right])<3for{E}^{″}\subseteq E\left(C_3\right)with{E}^{″}\ne \varnothing ,\hfill \\ G_i\cup G\left[E^{‴}\right]\hfill & ifc(G_i\cup G\left[E^{‴}\right])=3for{E}^{‴}\subseteq E\left(C_3\right)with{E}^{‴}\ne \varnothing .\hfill \end{array}\right.$$

If, for each $1\le i\le 3$, $c\left(H_i\right)=\rho$ with $\rho =2$ or 3, then $\{H_1,H_2,H_3\}$ is called a decomposition of equal cutwidth ρ of G; if there are at least two $H_i^{\prime }s$ (say $H_1,H_3$) such that $c\left(H_1\right)=2$ and $c\left(H_3\right)=3$, then $\{H_1,H_2,H_3\}$ is called a decomposition of nonequal cutwidth ρ with $\rho =2$ or 3 of G, where $E^{\prime },E^{″}$ and $E^{‴}$ are not necessarily distinct, and $E^{\prime }$ is not necessarily non-empty.

Lemma 15.

With notation in Definition 5, let G be 4-cutwidth critical with the central cycle $C_3=x_1{x}_2{x}_3{x}_1$, $x_1,x_2,x_3$ be cut vertices in G, and $C_3$ has at least two vertices (say $x_2,x_3$) such that $d_G\left(x_2\right)\ge 4$ and $d_G\left(x_3\right)\ge 4$. If $\{H_1,H_2,H_3\}$ is a decomposition of nonequal cutwidth ρ with $\rho =2$ or 3, then $H_i$ is ρ-cutwidth critical for $1\le i\le 3$ except $M_4$ in Figure 2.

Proof. 

Since $\{H_1,H_2,H_3\}$ is a decomposition of nonequal cutwidth $\rho$ with $\rho =2$ or 3, we can assume that $c\left(H_2\right)=c\left(H_3\right)=3$, but $c\left(H_1\right)=2$, implying $c(G_1\cup G\left[E\left(C_3\right)\right])\le 2$. Since G is 4-cutwidth critical with $d_G\left(x_2\right)\ge 4$ and $d_G\left(x_3\right)\ge 4$, $H_1=K_{1,3}$ or $C_3$, $H_i\ne {\tau }_2,{\tau }_4$ or ${\tau }_5$ for $i=2,3$ by Lemma 6, meaning to that $H_i={\tau }_1$ or ${\tau }_3$ for $i=2,3$. Thus, for $H_2$ and $H_3$, there are three cases to consider: (i) $H_2=G_2\cup G\left[\{x_2{x}_3,x_2{x}_1\}\right]=K_{1,5}$, $H_3=G_3\cup G\left[\{x_3{x}_2,x_3{x}_1\}\right]=K_{1,5}$; (ii) $H_2=G_2\cup G\left[\{x_2{x}_3,x_2{x}_1\}\right]=K_{1,5}$, $H_3=G_3\cup C_3={\tau }_3$; (iii) $H_2=G_2\cup C_3={\tau }_3$, $H_3=G_3\cup C_3={\tau }_3$, where $x_2,x_3$ are the central vertices of $H_2$ and $H_3$, respectively (see $M_5,M_6,M_7$ in Figure 2 and Figure 6d,e below). In any of Cases (i)–(iii), $H_1=G_1=K_{1,3}$ with $d_G\left(x_1\right)=3$ or $C_3$ with $d_G\left(x_1\right)=4$. Thus, we can see that $H_i$ is 2-cutwidth critical for $i=1$ and 3-cutwidth critical for $i=2,3$. Now let $c\left(H_3\right)=3$ but $c\left(H_1\right)=c\left(H_2\right)=2$ with $H_1=C_3$ and $H_2=C_3$; then, we can conclude that $G=M_4$ by the 4-cutwidth criticality of G, which has a decomposition $\{K_{1,3},C_3,C_3^{\prime },C_3^{″}\}$ of equal cutwidth two in which $C_3^{\prime }$ and $C_3^{″}$ are the copies of $C_3$. This is because in this case, if $\{K_{1,5},C_3^{\prime },C_3^{″}\}$ is a decomposition of nonequal cutwidth of 2 and 3 of G, then edge $x_1{x}_2\notin E\left(K_{1,5}\right)\cup E\left(C_3^{\prime }\right)\cup E\left(C_3^{″}\right)$. As $x_2{x}_3\notin E\left(H_i\right)$ for each $1\le i\le 3$ in this case, this decomposition of nonequal cutwidth does not hold. Thus, this case is not possible. The proof is complete. □

Lemma 16.

With notation in Definition 5, let $\{H_1,H_2,H_3\}$ be a decomposition of nonequal cutwidth ρ with $\rho =2$ or 3 of 4-cutwidth graph G with the central cycle $C_3=x_1{x}_2{x}_3{x}_1$. If $H_i$ is ρ-cutwidth critical for $1\le i\le 3$, then G is 4-cutwidth critical, where $x_1,x_2,x_3$ are all cut vertices, and $C_3$ has at least two vertices (say $x_2,x_3$) such that $d_G\left(x_2\right)\ge 4$ and $d_G\left(x_3\right)\ge 4$.

Proof. 

Let $\pi$ be an optimal labeling of G with $\pi \left(x_1\right)<\pi \left(x_2\right)<\pi \left(x_3\right)$ and intervals $I_1=[1,\pi \left(x_1\right)],I_2=(\pi \left(x_1\right),\pi \left(x_3\right)),I_3=[\pi \left(x_3\right),n]$ with $n=\left|V\right(G\left)\right|$, respectively. Then, $G_1$ is embedded in $I_1$ with congestion 3, $G_2$ is embedded in $I_2$ with congestion 4, $G_3$ is embedded in $I_3$ with congestion 3. Herein, $G_1$ and $G_3$ are a star $K_{1,3}$ with center $x_i$ or two stars $K_{1,3}$ with an identifying leaf at $x_i$$(i=1,3)$. Let $H_i$ denote $G_i$ combining with the two edges in $C_3$ incident with $v_i$. Then $H_i\in \{{\tau }_1,{\tau }_3\}$ for $i=1,3$. As for $G_2$ embedded in $I_2$ with congestion 4, since the central cycle $C_3$ yields congestion 2 in $I_2$, we chose $G_2$ as a 2-cutwidth critical tree, namely, a $K_{1,3}$, such that either $d_G\left(x_2\right)=3$ or $d_G\left(x_2\right)=5$. For this construction, the maximum congestion is 4, i.e., $c\left(G\right)=4$. Furthermore, for any edge $e\in E\left(G\right)$, if $e\in \{x_1{x}_2,x_1{x}_3,x_2{x}_3\}$, then the deletion of e reduces the congestion 2 of cycle-edge in $I_2$ by one. Hence $H_2$ embedded in $I_2$ has congestion 3, and so $c(G-e)<4$. If $e\notin \{x_1{x}_2,x_1{x}_3,x_2{x}_3\}$, for Case (i) in Proof of Lemma 15, two subcases need to be considered: $\left(a\right)$ $G_i=K_{1,3}$ with $d_G\left(x_i\right)=5$ for each $1\le i\le 3$; $\left(b\right)$ $G_i=K_{1,3}$ with $d_G\left(x_i\right)=5$ for $i=1,3$, but $G_2=C_3$ with $d_G\left(x_2\right)=4$ for $i=1,3$. Without loss of generality, we can let $e\in E\left(G_2\right)$ with $G_3=K_{1,3}$. Since $G_2-e=K_{1,3}-e$ with congestion 1, we can embed $G_1$ in $I_1$, $G_2-e$ in $(\pi \left(v_1\right),\pi \left(v_3\right)-1)$ and $G_3$ in $[\pi \left(v_3\right)-1,n-1]$, respectively, which results in $c(G-e)=3$. So does the case of $e\in E\left(G_1\right)$ (or $E\left(G_3\right)$). Likewise, for Cases (ii) and (iii) in Proof of Lemma 15, $c(G-e)=3$ for any $e\in E\left(G\right)$ also. Therefore, G is 4-cutwidth critical. The lemma holds. □

Lemma 17.

With notation in Definition 5, let G be 4-cutwidth critical with the central cycle $C_3=x_1{x}_2{x}_3{x}_1$, where $x_1,x_2,x_3$ are all cut vertices in G, and $C_3$ has at most one vertex (say $x_1)$ such that $d_G\left(x_1\right)\ge 4$. If $\{H_1,H_2,H_3\}$ is a decomposition of equal cutwidth 3, then $H_i$( or $H_i-x_i{x}_i^{\prime }$ with $x_i^{\prime }\in N_G\left(x_i\right)\cap V\left(G_i\right))$ is 3-cutwidth critical for $1\le i\le 3$.

Proof. 

We first give Claim 1 below.

Claim 1. There is at least $H_i$$(1\le i\le 3)$ such that $H_i$ is one of $G_i\cup G\left[\{x_i{x}_{i-1},x_i{x}_{i+1}\}\right]$ and $G_i\cup C_3$ (say $G_i\cup G\left[\{x_i{x}_{i-1},x_i{x}_{i+1}\}\right]$) with $c\left(H_i\right)=3$, where $x_0=x_3$ and $x_4=x_1$.

Let $H_i=G_i$ or $G_i+x_i{x}_{i+1}$ with $c\left(H_i\right)=3$ for each $1\le i\le 3$. As the arguments are similar, we only consider two cases: (a) $H_1=G_1$ with $d_G\left(x_1\right)\ge 4$, $H_i=G_i+x_i{x}_{i+1}$ with $d_G\left(x_i\right)=3$ for $i=2,3$; (b) $H_i=G_i+x_i{x}_{i+1}$ with $d_G\left(x_i\right)=3$ for each $1\le i\le 3$. For Case (a), $x_i{x}_{i+1}$ is a pendent edge of $H_i$ for $i=2,3$, $d_{H_2}\left(x_2\right)=2$ and $d_{H_3}\left(x_3\right)=2$. So, $H_2=G_2+x_2{x}_1$ also and $c\left(G_2\right)=c\left(G_3\right)=3$ by a series reduction in $H_2$ and $H_3$, respectively. Thus, $G-x_2{x}_3=P_3\circ (G_2,G_1,G_3)$ which results in that $c(G-x_2{x}_3)=4$ by Theorem 3, contrary to the criticality of G. For Case (b), $d_G\left(x_i\right)=3$ and $d_{H_i}\left(x_i\right)=2$ for each $1\le i\le 3$, so every $c\left(G_i\right)=c\left(H_i\right)=3$ by a series reduction in $H_i$ and $d_{G_i}\left(x_i\right)=1$. Thus, there is an edge in $C_3$, say $x_1{x}_3$, such that $G-x_1{x}_3=K_{1,3}\circ (G_1,G_2,G_3)$. Hence $c(G-x_1{x}_3)=4$ by Theorem 2, also a contradiction. Claim 1 holds.

From Claim 1 and assumption, there are nine cases to consider, as follows (see graphs $\left(a\right)$–$\left(c\right)$ in Figure 6 below):

(1)

$H_1=G_1\cup G\left[\{x_1{x}_2,x_1{x}_3\}\right]$ with $d_G\left(x_1\right)\ge 4$, $H_2=G_2$ and $H_3=G_3$;

(2)

$H_1=G_1\cup G\left[\{x_1{x}_2,x_1{x}_3\}\right]$ with $d_G\left(x_1\right)\ge 4$, $H_2=G_2\cup G\left[\{x_2{x}_1,x_2{x}_3\}\right]$ and $H_3=G_3$;

(3)

$H_1=G_1\cup G\left[\{x_1{x}_2,x_1{x}_3\}\right]$ with $d_G\left(x_1\right)\ge 4$, $H_2=G_2\cup G\left[\{x_2{x}_1,x_2{x}_3\}\right]$ and $H_3=G_3\cup G\left[\{x_3{x}_1,x_3{x}_2\}\right]$;

(4)

$H_1=G_1\cup C_3$ with $d_G\left(x_1\right)\ge 4$, $H_2=G_2$ and $H_3=G_3$;

(5)

$H_1=G_1\cup C_3$ with $d_G\left(x_1\right)\ge 4$, $H_2=G_2\cup G\left[\{x_2{x}_1,x_2{x}_3\}\right]$ and $H_3=G_3$;

(6)

$H_1=G_1\cup C_3$ with $d_G\left(x_1\right)\ge 4$, $H_2=G_2\cup G\left[\{x_2{x}_1,x_2{x}_3\}\right]$ and $H_3=$$G_3\cup G\left[\{x_3{x}_1,x_3{x}_2\}\right]$;

(7)

$H_1=G_1\cup G\left[\{x_1{x}_2,x_1{x}_3\}\right]$, $H_2=G_2$ and $H_3=G_3$;

(8)

$H_1=G_1\cup G\left[\{x_1{x}_2,x_1{x}_3\}\right]$, $H_2=G_2\cup G\left[\{x_2{x}_1,x_2{x}_3\}\right]$ and $H_3=G_3$;

(9)

$H_1=G_1\cup G\left[\{x_1{x}_2,x_1{x}_3\}\right]$, $H_2=G_2\cup G\left[\{x_2{x}_1,x_2{x}_3\}\right]$ and $H_3=$$G_3\cup G\left[\{x_3{x}_1,x_3{x}_2\}\right]$,

where $d_G\left(x_i\right)=3$ for $i=2,3$ in Cases $\left(1\right)$–$\left(6\right)$, and $d_G\left(x_i\right)=3$ for each $1\le i\le 3$ in Cases $\left(7\right)$–$\left(9\right)$. We consider Case $\left(1\right)$ by contradiction. Assuming that there is at least an edge $xy\in E\left(H_i\right)$ such that $c(H_i-xy)=3$, i.e., $H_i$ is not 3-cutwidth critical. There are three subcases to consider: $\left(i\right)$ $c(H_1-xy)=3$ with $xy\in E\left(H_1\right)$; $\left(ii\right)$ $c(H_2-xy)=3$ with $xy\in E\left(H_2\right)$; $\left(iii\right)$ $c(H_3-xy)=3$ with $xy\in E\left(H_3\right)$. For Subcase $\left(i\right)$, by assumption and Definition 3, for $i=2,3$, $d_G\left(x_2\right)=d_G\left(x_3\right)=3$, $G_i$ is 3-cutwidth critical, and $c((G_i-x^{\prime }{y}^{\prime })\cup G\left[E^{\prime }\right])<3for{x}^{\prime }{y}^{\prime }\in E\left(G_i\right)and{E}^{\prime }\subseteq E\left(C_3\right)with{E}^{\prime }\ne \varnothing$, so either $G_i\in \{{\tau }_1,{\tau }_4\}$ or $G_i=K_2\cup {\tau }_5$. Thus, if $xy\in \{x_1{x}_2,x_1{x}_3\}$ (say $xy=x_1{x}_2$), then $G-xy$ is changed to ${\oplus }_{x_3}(H_1-xy,G_2,G_3)$ with cutwidth 4 resulting in $c(G-xy)=4$; if $xy\notin \{x_1{x}_2,x_1{x}_3\}$, i.e., $xy\in E\left(G_1\right)$ then $G-xy-x_2{x}_3$ is changed to be ${\oplus }_{x_2}(H_1-xy,G_2,G_3)$ with cutwidth 4 resulting in $c(G-xy)\ge c(G-xy-x_2{x}_3)=4$. So $c(G-xy)=4$ by $c(G-xy)\le c\left(G\right)=4$ again, and contrary to that, G is 4-cutwidth critical. For Subcase $\left(ii\right)$, we can conclude that $H_1=K_{1,5}$ and either $G_3\in \{{\tau }_1,{\tau }_4\}$ or $G_i=K_2\cup {\tau }_5$ with cutwidth 3. By Lemma 1(3), an optimal labeling $f^{*}$ by the order $(V\left(H_2\right)-xy),V(H_1+x_2{x}_3),V\left(H_3\right))$ of $G-xy$ can be obtained, and $c(G-xy,f)=4$, implying $c(G-xy)\le 4$. So, $c(G-xy)=4$ by the optimality of $f^{*}$, also a contradiction. The argument of Subcase $\left(iii\right)$ is the same as that of Subcase $\left(ii\right)$, omitted here. Thus, for Case $\left(1\right)$, ${\overline{G}}_i$ is 3-cutwidth critical for $1\le i\le 3$. Similarly, for Cases $\left(2\right)$–$\left(9\right)$, $H_i$ is also 3-cutwidth critical for $1\le i\le 3$. This completes the proof. □

Lemma 18.

With notation in Definition 5, let $\{H_1,H_2,H_3\}$ be a decomposition of equal cutwidth 3 of graph G with the central cycle $C_3=x_1{x}_2{x}_3{x}_1$, where $x_1,x_2,x_3$ are all cut vertices of G, and $C_3$ has at most one vertex (say $x_1)$ such that $d_G\left(x_1\right)\ge 4$, and either $\{x_1{x}_2,x_1{x}_3\}\subset E\left(H_1\right)$ or $E\left(C_3\right)\subset E\left(H_1\right)$. If $H_i$ is 3-cutwidth critical or there are at least a $H_i=G_i=x_i{x}_i^{\prime }+{\tau }_5$ with $x_i^{\prime }\in N_G\left(x_i\right)$ for $1\le i\le 3$, then G is 4-cutwidth critical.

Proof. 

By Lemmas 1$\left(3\right)$, we can show $c\left(G\right)=4$. By assumption again, $H_1\in \{{\tau }_1,{\tau }_3\}$ and $d_G\left(x_2\right)=d_G\left(x_3\right)=3$. There are nine cases $\left(1\right)$–$\left(9\right)$ listed in Proof of Lemma 17 to consider. For each case $\left(i\right)$ $(1\le i\le 9)$, via using an argument similar to that of Lemma 16, we can show $c\left(G^{\prime }\right)\le 3$ for any $G^{\prime }\in \mathcal{M}$$\left(G\right)$, omitted here. □

Lemma 19.

Let G be a 2-connected graph with a central cycle $C_3=x_1{x}_2{x}_3{x}_1$. Then G is 4-cutwidth critical with a decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ of equal cutwidth 3 if and only if $G=M_8$ (see Figure 2).

Proof. 

Sufficiency. Since $G=M_8$, G is 4-cutwidth critical by Lemma 6. Clearly, let ${\overline{G}}_i=G\left[\{x_1,x_2,x_3,y_i\}\right]$ for $1\le i\le 3$, then $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ is a decomposition desired because of ${\overline{G}}_i={\tau }_5$ for each $1\le i\le 3$.

Necessity. In fact, since G is 2-connected with a central cycle $C_3=x_1{x}_2{x}_3{x}_1$ and a decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$ of equal cutwidth 3, the arbitrary two vertices $x_i$ and $x_{i+1}$$(1\le i\le 3)$ of $C_3$ must be in a cycle $C_t^{\prime }$$(t\ge 3)$ and $C_t^{\prime }\ne C_3$, where $x_4=x_1$. That is to say, by the criticality of G, there must be another vertex $y_i\ne x_i$ in G such that $y_i{x}_i\in E\left(G\right)$ and $y_i{x}_{i+1}\in E\left(G\right)$ for each $1\le i\le 3$. In this case, $G=M_8$, induced by $\{x_1,x_2,x_3,y_1,y_2,y_3\}$. Hence $G=M_8$. □

Lemma 20.

Assume that G is a 4-cutwidth critical graph with a central cycle $C_3=x_1{x}_2{x}_3{x}_1$, then G has an edge-disjoint decomposition $\{G_1,G_2,G_3,C_3\}$ of equal cutwidth 2 if and only if $G\in \{M_3,M_4,M_5,M_6,M_7\}$ (see Figure 2), where $x_i$ is a cut vertex and $G_i$ is the connected component of $G-E\left(C_3\right)$ leading from $x_i$ for $1\le i\le 3$.

Proof. 

Sufficiency is obvious in Lemma 6, omitted here.

Necessity: Let $\pi$ be an optimal labeling of G with $\pi \left(x_1\right)<\pi \left(x_2\right)<\pi \left(x_3\right)$ and $\left|V\right(G\left)\right|=n$. Then, the number set $\mathcal{S}$${}_n$ is divided into three intervals $I_1=[1,\pi \left(x_1\right)],I_2=(\pi \left(x_1\right),\pi \left(x_2\right))$ and $I_3=[\pi \left(x_3\right),n]$ and $G_1,G_2,G_3$ are embedded into $I_1,I_2,I_3$ in different manners, respectively. As $G_1,G_2,G_3$ are all 2-cutwidth graphs and $x_i$ is a cut vertex in G for $1\le i\le 3$, $G_1$ is embedded into $I_1$ with congestion 2, $G_2$ is embedded into $I_2$ with congestion 4, and $G_3$ is embedded into $I_3$ with congestion 2. By the criticality of G and $c\left(K_{1,3}\right)=c\left(C_3\right)=2$, $G_i$ is either a star $K_{1,3}$ with the 3-degree vertex $x_i$ or a cycle $C_3^{\left(i\right)}$, which is a copy of $C_3$ for $i=1,3$. As for $G_2$ embedded in $I_2$ with a congestion of 4, the central cycle $C_3$ leads to a congestion of 2 in $I_2$, so $G_2$ must be either a $K_{1,3}$ or a copy $C_3^{\left(2\right)}$ of $C_3$ such that $d_G\left(x_2\right)=3,4$ or 5. Thus, G must be one member of $\{M_3,M_4,M_5,M_6,M_7\}$, each element of which has a edge-disjoint decomposition $\{G_1,G_2,G_3,C_3\}$ of equal cutwidth 2, where $G_i$ is either $K_{1,3}$ or $C_3$ for $1\le i\le 3$. □

Theorem 8.

For a 4-cutwidth nontree graph G with a central cycle $C_3=x_1{x}_2{x}_3{x}_1$, G is 4-cutwidth critical if and only if G has one of the following configurations.

(1)

G has a decomposition $\{H_1,H_2,H_3\}$ of nonequal cutwidth ρ with ρ=2 or 3, each of which is ρ-cutwidth critical, where $x_i$ is a cut vertex for each $1\le i\le 3$ and there are at least two vertices (say $x_2,x_3$) such that $d_G\left(x_2\right)\ge 4$ and $d_G\left(x_3\right)\ge 4$ (see $M_5$–$M_7$ in Figure 2 and Illustration in Figure 6d,e);

(2)

G has a decomposition $\{H_1,H_2,H_3\}$ of equal cutwidth 3 in which $H_i$ or $H_i-x_i{x}_i^{\prime }$ with $x_i^{\prime }\in N_G\left(x_i\right)\cap V\left(G_i\right)$ is 3-cutwidth critical, and at least a $H_i$ (say $H_1$) contains at least two edges $x_1{x}_2$ and $x_1{x}_3$ of $C_3$, where $x_i$ is a cut vertex for each $1\le i\le 3$ and there is at most a vertex (say $x_3$) such that $d_G\left(x_3\right)\ge 4$ (see Illustration in Figure 6a–c);

(3)

G is 2-connected and $G=M_8$ (see Figure 2) with a decomposition $\{H_1,H_2,H_3\}$ of equal cutwidth 3 in which $H_i=G\left[\{x_1,x_2,x_3,y_i\}\right]={\tau }_5$ for $1\le i\le 3$;

(4)

$G\in \{M_3,M_4,M_5,M_6,M_7\}$ with an edge-disjoint decomposition $\{G_1,G_2,G_3,C_3\}$ of equal cutwidth 2, in which $G_i$ is either $K_{1,3}$ or a copy $C_3^{\prime }$ of $C_3$ for $1\le i\le 3$ (see $M_3$–$M_7$ in Figure 2).

Figure 6. Illustrations of Theorem 8.

Figure 6. Illustrations of Theorem 8.

In Figure 6a–c, for $i=1,2,3$, $G_i={\tau }_1,{\tau }_4,{\tau }_5+x_i{x}_i^{\prime }$ with $x_i^{\prime }\in V\left({\tau }_5\right)$ or $G_i+x_i{x}_{i-1}+x_i{x}_{i+1}={\tau }_2,{\tau }_3$ with $x_0=x_3$, but in Figure 6c, there is at least a $G_i$, say $G_3$, such that $G_3+x_3{x}_1+x_3{x}_2={\tau }_2$ or ${\tau }_3$. In Figure 6d,e, if $G_1=K_{1,3}$ then $d_G\left(x_1\right)=3$, i.e., $x_1$ is a pendant vertex of $K_{1,3}$ in this case, $C_3^{\prime }$ is a copy of $C_3$.

4.2. Graphs with a Central Cycle of Length Four

For a graph G with a central cycle $C_4=x_1{x}_2{x}_3{x}_4{x}_1$ with length 4, suppose that $G_i$$(1\le i\le 4)$ is the ith connected component leading from $x_i$ of $G-E\left(C_4\right)$, $c\left(G_1\right)\ge c\left(G_2\right)\ge c\left(G_4\right)\ge c\left(G_3\right)$ and $d_G\left(x_1\right)\ge 4$, and $G-E\left(C_4\right)$ has no $G_i$, such that $c\left(G_i\right)=3$ but $c(G_i^{\prime }\cup \{x_i{x}_{i-1},x_i{x}_{i+1}\})=3$ for any proper subgraph $G_i^{\prime }\subset G_i$. Let ${\overline{G}}_1=G_1$ when $c\left(G_1\right)=3$ or $G_1\cup G\left[E^{\prime }\right]$ with $E^{\prime }\subseteq E\left(C_4\right)$ and $E^{\prime }\ne \varnothing$ when $c\left(G_1\right)<3$, ${\overline{G}}_2=G_2\cup G_3\cup (C_4-x_1{x}_4+\{x_1{x}_1^{\prime },x_1{x}_1^{″}\})$ and ${\overline{G}}_4=G_4\cup G_3\cup (C_4-x_1{x}_2+\{x_1{x}_1^{\prime },x_1{x}_1^{″}\})$ for $x_1^{\prime },x_1^{″}\in V\left(G_1\right)$ with $x_1^{\prime }\ne x_1^{″}$, where $x_1,x_2,x_3,x_4$ are all cut vertices in G, and there is at least a vertex between $x_2$ and $x_4$ (say $x_4$) such that $d_G\left(x_2\right)\ge 4$. Then, we have the following:

Lemma 21.

For a graph G with the central cycle $C_4=x_1{x}_2{x}_3{x}_4{x}_1$, if $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_4\}$ is a decomposition of equal cutwidth 3 of G and ${\overline{G}}_i$ is 3-cutwidth critical for each $i\in \{1,2,4\}$, then G is 4-cutwidth critical (see Illustrations in Figure 7).

Proof. 

By assumption, $d_G\left(x_2\right)=d_G\left(x_3\right)=3$, and since ${\overline{G}}_i$ is 3-cutwidth critical for $i\in \{1,2,4\}$, ${\overline{G}}_1\in \{{\tau }_1,{\tau }_3,{\tau }_4,{\tau }_5\}$, ${\overline{G}}_2={\tau }_2$ with ${\overline{G}}_2=K_{1,3}$ and ${\overline{G}}_4={\tau }_2$ with $G_2=K_{1,3}$ or ${\tau }_3$ with $G_3=C_3$ resulting in $G_3=K_2$. Suppose that $\pi :V\left(G\right)\to \mathcal{S}$${}_n$ is a labeling of G with $\pi \left(x_1\right)<\pi \left(x_2\right)<\pi \left(x_3\right)<\pi \left(x_4\right)$, then $\mathcal{S}$${}_n$ is partitioned into three intervals $I_1=[1,\pi \left(x_1\right)],I_2=(\pi \left(x_2\right),\pi \left(x_4\right)]$ and $I_3=(\pi \left(x_4\right),n]$. Now, we embed $G_1$ in $I_1$ with congestion 3, ${\overline{G}}_2-\{x_1{x}_1^{\prime },x_1{x}_1^{″}\}$ in $I_2$ and connect $x_1{x}_4$ with congestion 4, $G_4-x_4$ in $I_3$ with congestion 2. Thus, $c(G,\pi )=4$, implying $c\left(G\right)\le 4$. On the other hand, $c\left(G\right)\ge 4$. Hence $c\left(G\right)=4$.

The remaining is to show $c(G-e)<4$ for any $e\in E\left(G\right)$. There are three cases to consider: $\left(1\right)$ $e\in E\left(G_1\right)$; $\left(2\right)$ $e\in E\left(C_4\right)$; $\left(3\right)$e is a pendant edge of $G_i$ for $i=2,3,4$. For Case $\left(1\right)$, $c(G_1-e)\le 2$. Since $d_G\left(x_2\right)=3$, by Lemma 1(3), if $e=v_1{v}_2$ is a pendant edge of $G_1$ with $d_G\left(v_2\right)=1$, then we can find an optimal labeling ${\pi }^{\prime }:V(G-v_2)\to \mathcal{S}$${}_{n-1}$ with $c(G-v_2,{\pi }^{\prime })=3$, under which $G_2-x_2$ is embedded in interval $[1,min\{\pi \left(v\right):v\in V(G_1-v_2)\})$ with congestion 3. If $e\in E\left(C_3\right)$ (note that ${\overline{G}}_1={\tau }_4$ or ${\tau }_5$ in this subcase), then we can find an optimal labeling ${\pi }^{″}:V(G-e)\to \mathcal{S}$${}_n$ with $c(G-e,{\pi }^{″})=3$, under which $G_2-x_2$ is embedded in interval $[1,min\{\pi \left(v\right):v\in V\left(G_1\right)\}]$ with congestion 3. So $c(G-e)=3$. Similarly, for Cases $\left(2\right)$ and $\left(3\right)$, $c(G-e)=3$ also. Hence, G is 4-cutwidth critical. □

Lemma 22.

Let G be a 4-cutwidth critical graph with the central cycle $C_4=x_1{x}_2{x}_3{x}_4{x}_1$. If G has a decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_4\}$ of equal cutwidth 3, then ${\overline{G}}_i$ is 3-cutwidth critical for each $i\in \{1,2,4\}$ (see Illustrations in Figure 7).

Proof. 

By contradiction, suppose that there is at least a ${\overline{G}}_i$ (say ${\overline{G}}_2$) such that ${\overline{G}}_2$ is not 3-cutwidth critical, then there exists an edge $e\in E\left({\overline{G}}_2\right)$ such that $c({\overline{G}}_2-e)=3$ also. Two cases need to be considered: $\left(1\right)$ $e=v{v}^{\prime }$ is a pendant edge with $d_G\left(v^{\prime }\right)=1$ in ${\overline{G}}_2$; $\left(2\right)$ $e\in E\left(C^{\prime }\right)$ if ${\overline{G}}_2$ contains a cycle $C^{\prime }$ which does not equal the central cycle $C_4$. Using an argument similar to that of Lemma 21, for Case $\left(1\right)$, we can find a labeling $\pi :V(G-v^{\prime })\to \mathcal{S}$${}_{n-1}$ with $c(G-v^{\prime })=4$, thereby contradicting that G is 4-cutwidth critical. Furthermore, likewise, for Case $\left(2\right)$, we can find a labeling $\pi :V(G-e)\to \mathcal{S}$${}_n$ with $c(G-e)=4$, also contradicting that G is 4-cutwidth critical. Similarly, if $e\in E\left({\overline{G}}_i\right)$ for $i=1$ or 4 then we can also find a contradiction to the assertion that G is 4-cutwidth critical. Therefore, ${\overline{G}}_i$ is 3-cutwidth critical for each $i\in \{1,2,4\}$. □

From Lemmas 21 and 22, the structure of a 4-cutwidth critical graph G with a central cycle $C_4=x_1{x}_2{x}_3{x}_4{x}_1$ can be obtained below.

Theorem 9.

Assume that G is a 4-cutwidth graph with a central cycle $C_4=x_1{x}_2{x}_3{x}_4{x}_1$, and $x_i$ is a cut vertex for $1\le i\le 4$, then G is 4-cutwidth critical if and only if G has a decomposition $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_4\}$ of equal cutwidth 3, each of which is 3-cutwidth critical, where ${\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_4$ are one of the following:

(1)

${\overline{G}}_1=K_{1,5}$ with the central vertex $x_1$ of $d_G\left(x_1\right)=5$ or ${\tau }_5$ with $d_G\left(x_1\right)=4$, and ${\overline{G}}_2$ and ${\overline{G}}_4$ are both in $\{{\tau }_2,{\tau }_3\}$, but ${\overline{G}}_2$ and ${\overline{G}}_4$ do not equal ${\tau }_3$ simultaneously (see Illustration in Figure 7a);

(2)

${\overline{G}}_1$ is homeomorphic to ${\tau }_3$ with the central vertex $x_1$ of $d_G\left(x_1\right)=4$ and $C_4\subset {\overline{G}}_1$, ${\overline{G}}_2$ and ${\overline{G}}_4$ are both in $\{{\tau }_2,{\tau }_3\}$. ${\overline{G}}_2,{\overline{G}}_4$ are not necessarily different (see Illustration in Figure 7b).

Figure 7. Illustrations of Theorem 9.

Figure 7. Illustrations of Theorem 9.

4.3. Graphs with a Central Cycle of Length at Least Five

Suppose that G is a graph with the central cycle $C_5=x_1{x}_2{x}_3{x}_4{x}_5{x}_1$, and for $1\le i\le 5$, $G-E\left(C_5\right)$ has no component $G_i$ leading from $x_i$, such that $c\left(G_i\right)=3$, but $c(G_i^{\prime }\cup \{x_i{x}_{i-1},x_i{x}_{i+1}\})=3$ for any proper subgraph $G_i^{\prime }\subset G_i$, $C_5$ has at most two $x_i^{\prime }s$ with $d_G\left(x_i\right)\ge 4$, where $x_0=x_5,x_6=x_1$. Let one of the following hold:

(1)

${\overline{G}}_1=G_1\cup G_2\cup G_5\cup (C_5-x_3{x}_4)$, ${\overline{G}}_i=G_i$ or $G_i+x_3{x}_4$ if $c\left(G_i\right)=3$ or $G_i+x_i{x}_{i-1}+x_i{x}_{i+1}$ if $c\left(G_i\right)<3$ for $i=3,4$ with $d_G\left(x_3\right)=d_G\left(x_4\right)=3$;

(2)

${\overline{G}}_1=G_1\cup G_2\cup G_5\cup (C_5-x_3{x}_4)$, ${\overline{G}}_3=G_3\cup (C_5-x_1{x}_5+x_2{x}_2^{\prime }+x_4{x}_4^{\prime })$, ${\overline{G}}_4=G_4\cup (C_5-x_1{x}_2+x_3{x}_3^{\prime }+x_5{x}_5^{\prime })$ with $d_G\left(x_3\right)=d_G\left(x_4\right)=4$ and $c\left(G_3\right)=c\left(G_4\right)=2$, $x_i^{\prime }\in N_G\left(x_i\right)\cap V(G_i-x_i)$ for $2\le i\le 5$;

(3)

${\overline{G}}_1$ is homeomorphic to subgraph $(G_1+x_1{x}_2+x_1{x}_5)\cup G_2\cup G_5$, ${\overline{G}}_3=G_3\cup (C_5-x_1{x}_5+x_2{x}_2^{\prime }+x_4{x}_4^{\prime })$, ${\overline{G}}_4=G_4\cup (C_5-x_1{x}_2+x_3{x}_3^{\prime }+x_5{x}_5^{\prime })$ with $c\left(G_3\right)=c\left(G_4\right)=2$, where $C_5$ has at most two 4-degree vertices (say, $x_1$ and $x_4$) which are nonadjacent.

Then, we have the following:

Lemma 23.

For a graph G with the central cycle $C_5=x_1{x}_2{x}_3{x}_4{x}_5{x}_1$, if G is 4-cutwidth critical and $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_4\}$ is a subgraph decomposition of equal cutwidth 3 of G, then ${\overline{G}}_i$(or ${\overline{G}}_i-x_i)$ is 3-cutwidth critical for $i\in \{1,3,4\}$ (see Illustrations in Figure 8).

Proof. 

By contradiction, we first consider Case $\left(1\right)$ above. Suppose that there exists some ${\overline{G}}_i$, say ${\overline{G}}_1$ first, such that ${\overline{G}}_1$ is not 3-cutwidth critical. There are two subcases to consider: (i) ${\overline{G}}_1$ contains no cycle; (ii) ${\overline{G}}_1$ contains at least a cycle. For (i), ${\overline{G}}_1$ has at least a pendant vertex v such that $c({\overline{G}}_1-v)=3$. By $d_G\left(x_3\right)=d_G\left(x_4\right)=3$, let $x_3{x}_3^{\prime },x_4{x}_4^{\prime }$ be cut edges in G with $x_3^{\prime }\in V(G_3-x_3)\cap N_G\left(x_3\right)$ and $x_4^{\prime }\in V(G_4-x_4)\cap N_G\left(x_4\right)$. Then $d_{G-v}\left(x_3\right)=d_{G-v}\left(x_4\right)=3$, and $x_3{x}_3^{\prime },x_4{x}_4^{\prime }$ are both cut edges in $G-v$ clearly. So, by Lemma 1(3), $G-v$ has an optimal labeling $\pi$ such that the vertices in each of $V(G_3-x_3)$, $V({\overline{G}}_1-v+x_3{x}_4)$ and $V(G_4-x_4)$ are labeled consecutively. Without loss of generality, let $max\{\pi \left(v\right):v\in V(G_3-x_3)\}<min\{\pi \left(v\right):v\in V({\overline{G}}_1-v+x_3{x}_4)\}$ and $max\{\pi \left(v\right):v\in V({\overline{G}}_1-v+x_3{x}_4)\}<min\{\pi \left(v\right):v\in V(G_4-x_4)\}$. Then $c(G-v,\pi )=c({\overline{G}}_1-v)+1=4$. Since $\pi$ is optimal, $c(G-v)=c(G-v,\pi )=4$, contradicting that G is 4-cutwidth critical. For (ii), two subcases need to be considered: (a) ${\overline{G}}_1$ has at least a pendant vertex v such that $c({\overline{G}}_1-v)=3$; (b) ${\overline{G}}_1$ has at least a non-pendant edge e such that $c({\overline{G}}_1-e)=3$. Subcase (a) is the same as case (i), omitted here; For subcase (b), using a similar method to that of case (i), we can show $c(G-e)=4$, also a contradiction. Now, we consider ${\overline{G}}_3$ or ${\overline{G}}_4$, and without loss of generality; let $c({\overline{G}}_3-x_3{x}_3^{\prime })=c(G_3-x_3{x}_3^{\prime })=3$ with $x_3^{\prime }\in N_G\left(x_3\right)\cap V(G_3-x_3)$ and ${\overline{G}}_4=G_4+x_4{x}_3+x_4{x}_5$. Assume that there is an edge e such that $c({\overline{G}}_3-x_3{x}_3^{\prime }-e)=3$, i.e., ${\overline{G}}_3-x_3{x}_3^{\prime }-e$ is not 3-cutwidth critical. Similar to Case (i), $d_{G-e}\left(x_3\right)=d_{G-e}\left(x_4\right)=3$, and $x_3{x}_3^{\prime },x_4{x}_4^{\prime }$ are both cut edges in $G-e$. By Lemma 1(3), $G-e$ has an optimal labeling ${\pi }^{\prime }$ such that the vertices in each of $V({\overline{G}}_1+x_3{x}_4)$, $V(G_3-x_3-e)$ and $V(G_4-x_4)$ are labeled consecutively with $max\{{\pi }^{\prime }\left(v\right):v\in V({\overline{G}}_1+x_3{x}_4)\}<min\{{\pi }^{\prime }\left(v\right):v\in V(G_3-x_3-e)\}$ and $max\{{\pi }^{\prime }\left(v\right):v\in V(G_3-x_3-e)\}<min\{{\pi }^{\prime }\left(v\right):v\in V(G_4-x_4)\}$. Thus $c(G-e)=c(G-e,{\pi }^{\prime })=c(G_3-x_3{x}_3^{\prime }-e)+1=4$, contradicting that G is 4-cutwidth critical. Likewise, let ${\overline{G}}_3$ and ${\overline{G}}_4$ be one of the followings, and one of $\{{\overline{G}}_3,{\overline{G}}_4\}$ be not 3-cutwidth critical: (A1) each ${\overline{G}}_i=G_i$ with $c({\overline{G}}_i-x_i{x}_i^{\prime })=c(G_i-x_i{x}_i^{\prime })=3$ for $i=3,4$;

(A2) each ${\overline{G}}_i=G_i+x_i{x}_{i-1}+x_i{x}_{i+1}$ with $c\left(G_i\right)<3$ for $i=3,4$;

(A3) ${\overline{G}}_3=G_3$ with $c\left(G_3\right)=3$ but $c(G_3-x_3{x}_3^{\prime })<3$, ${\overline{G}}_4=G_4+x_4{x}_3+x_4{x}_5$ with $c\left(G_4\right)<3$;

(A4) ${\overline{G}}_3=G_3$ with $c\left(G_3\right)=3$, ${\overline{G}}_4=G_4$ with $c\left(G_4\right)=3$ but $c(G_4-x_4{x}_4^{\prime })<3$.

Then we can also obtain a contradiction to the assertion that G is 4-cutwidth critical. Hence, each ${\overline{G}}_i$(or ${\overline{G}}_i-x_i)$ is 3-cutwidth critical.

Similarly, for Cases $\left(2\right)$ and $\left(3\right)$ above, ${\overline{G}}_i$(or ${\overline{G}}_i-x_i)$ is also 3-cutwidth critical for $i\in \{1,3,4\}$. This completes the proof. □

Lemma 24.

For a 4-cutwidth graph G with the central cycle $C_5=x_1{x}_2{x}_3{x}_4{x}_5{x}_1$, if $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_4\}$ is a decomposition of equal cutwidth 3 of G, ${\overline{G}}_i$(or ${\overline{G}}_i-x_i)$ is 3-cutwidth critical for $i\in \{1,3,4\}$, then G is 4-cutwidth critical.

Proof. 

Three cases similar to those of Lemma 23 need to be considered. We first consider Case $\left(1\right)$ by contradiction. Suppose that G is not 4-cutwidth critical, i.e., there exists a pendant vertex v (or a non-pendant edge e) such that $c(G-v)=4$ (or $c(G-e)=4$). There are three subcases to consider: $\left(i\right)$ $v\in V\left({\overline{G}}_1\right)$ (or $e\in E\left({\overline{G}}_1\right)$); $\left(ii\right)$ $v\in V\left({\overline{G}}_2\right)$ (or $e\in E\left({\overline{G}}_2\right)$); $\left(iii\right)$ $v\in V\left({\overline{G}}_3\right)$ (or $e\in E\left({\overline{G}}_3\right)$). For Case $\left(i\right)$, by assumption, $c({\overline{G}}_1-v)<3$ (or $c({\overline{G}}_1-e)<3$). Since $d_G\left(x_3\right)=d_G\left(x_4\right)=3$, using a similar method to that of Lemma 22, we can verify that $c(G-v)<4$ (or $c(G-e)<4$) contrary to $c(G-v)=4$ (or $c(G-e)=4$). So, G is 4-cutwidth critical. Likewise, for Subcases $\left(ii\right)$ and $\left(iii\right)$, G is 4-cutwidth critical also.

Similarly, for Cases $\left(2\right)$ and $\left(3\right)$, G is 4-cutwidth critical also. This proof is completed. □

From Lemmas 23 and 24:

Theorem 10.

Assume that G is a 4-cutwidth graph with a central cycle $C_4=x_1{x}_2{x}_3{x}_4{x}_5{x}_1$, and $x_i$ is a cut vertex for $1\le i\le 5$, then G is 4-cutwidth critical if and only if G has a decomposition $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_4\}$ (or $\{{\overline{G}}_1,{\overline{G}}_3+x_3{x}_4,{\overline{G}}_4\}$, $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_4+x_4{x}_3\}$, $\{{\overline{G}}_1,{\overline{G}}_3+x_3{x}_4,{\overline{G}}_4+x_4{x}_3\})$ of equal cutwidth 3, where ${\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_4$ are one of the following:

(1)

${\overline{G}}_1\in \{{\tau }_2,{\tau }_3\}$ with the central vertex $x_1$ of degree three or four, for $i=3,4$, ${\overline{G}}_i$(or ${\overline{G}}_i-x_i)$ is one of $\{{\tau }_i:1\le i\le 5\}$ and $x_i$ satisfies: $\left(i\right)$ $d_G\left(x_i\right)=3$, $\left(ii\right)$ $x_i$ is not the central vertex of ${\overline{G}}_i$ when ${\overline{G}}_i\in \{{\tau }_1,{\tau }_2,{\tau }_3\}$, and $\left(iii\right)$ $x_i{x}_{i-1},x_i{x}_{i+1}$ are the pendant edges of ${\overline{G}}_i$ when ${\overline{G}}_i$ is ${\tau }_2$ or ${\tau }_3$ (see Illustration in Figure 8a);

(2)

${\overline{G}}_1$ is homeomorphic to ${\tau }_2$ with the central vertex $x_1$ of degree three, for $i=3,4$, ${\overline{G}}_i$ is homeomorphic to ${\tau }_2$ or ${\tau }_3$ with $G_i\in \{K_{1,3},C_3\}$, where $G_3,G_4$ are not necessarily different (see Illustration in Figure 8b);

(3)

${\overline{G}}_1$ is homeomorphic to ${\tau }_3$ with the central vertex $x_1$ of degree four, for $i=3,4$, ${\overline{G}}_i$ is homeomorphic to ${\tau }_2$ or ${\tau }_3$ with $G_i\in \{K_{1,3},C_3\}$, but if $G_3=C_3$, then $G_4\ne C_3$ and vice versa (see Illustration in Figure 8b).

Figure 8. Illustrations of Theorem 10.

Figure 8. Illustrations of Theorem 10.

In Figure 8a, ${\overline{G}}_i$ is either $G_i$ which is ${\tau }_1$ or ${\tau }_4$ or ${\tau }_5+x_i{x}_i^{\prime }$ with $x_i^{\prime }\in V\left({\tau }_5\right)$ or $G_i+x_i{x}_{i-1}+x_i{x}_{i+1}$ which is in $\{{\tau }_2,{\tau }_3\}$ for $i=3,4$; Additionally, if $G_1=K_{1,3}$ then $G_3,G_4$ can be 3-cycle $C_3$ simultaneously.

Lemma 25.

For a graph G with a central cycle $C_6=x_1{x}_2{x}_3{x}_4{x}_5{x}_6{x}_1$, G is 4-cutwidth critical if and only if $G\in \{M_{18},M_{19},M_{20}\}$ in Figure 9.

Proof. 

Sufficiency. For any $G\in \{M_{18},M_{19},M_{20}\}$, G can be easily shown to be 4-cutwidth critical by proving two conclusions: (1) $c\left(G\right)=4$; (2) $c\left(G^{\prime }\right)=3$ for any $G^{\prime }\in \mathcal{M}$$\left(G\right)$, omitted here.

Necessity. Let G be a 4-cutwidth critical graph with the central cycle $C_6=x_1{x}_2{x}_3{x}_{4}x$${}_5{x}_6{x}_1$.

Observation. For any $18\le i\le 20$, $M_i$ has a decomposition $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_5\}$ of equal cutwidth 3, where ${\overline{G}}_l=H_{l-1}\cup H_l\cup H_{l+1}={\tau }_2$ or ${\tau }_3$ with $H_l=G_l+x_l{x}_{l-1}+x_l{x}_{l+1}$ for $l\in \{1,3,5\}$, $H_0=H_6$ and $H_7=H_1$.

By observation, suppose by contradiction that $G\notin \{M_{18},M_{19},M_{20}\}$, then two cases need to be considered as follows.

Case 1. G has a decomposition $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_5\}$ of equal cutwidth 3, but there is at least an element in $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_5\}$, say ${\overline{G}}_3(=H_2\cup H_3\cup H_4)$, such that ${\overline{G}}_3$ does not equal ${\tau }_2$ (or ${\tau }_3$). In this case, $G-E\left(C_r\right)$ has at least a connected component $G_i$ leading from $x_i$, say $G_3$, such that $G_3\supset K_{1,3}$ (or $K_3$); this is because the connected component leading from $x_3$ in $M_{18}$ is $K_{1,3}$ (or in any of $\{M_{19},M_{20}\}$ is $K_3$. Without loss of generality, let $G_3$ be a minimum graph such that $K_{1,3}\subset G_3$ (or $K_3\subset G_3$), i.e., $|E\left(G_3\right)\backslash E\left(K_{1,3}\right)|=1$ (or $|E\left(G_3\right)\backslash E\left(K_3\right)|=1$). Then, by direct computations, $c\left(G\right)=4$ and $c\left({\overline{G}}_3\right)=3$, but G is not 4-cutwidth critical. Similarly, if $G_2\ne K_2$ or $G_4\ne K_2$ in ${\overline{G}}_3$ then G is not 4-cutwidth critical also. So this case is not possible.

Case 2. G has not a decomposition $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_5\}$ of equal cutwidth 3. In this case, there are at least an element in $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_5\}$, say ${\overline{G}}_1$, such that $c\left({\overline{G}}_1\right)$ is either at most 2 or at least 4, i.e., either $c\left({\overline{G}}_1\right)\le 2$ or $c\left({\overline{G}}_1\right)\ge 4$. Since G is 4-cutwidth critical, the subcase of $c\left({\overline{G}}_1\right)\ge 4$ is impossible. For the subcase of $c\left({\overline{G}}_1\right)\le 2$, we claim that $G_1$ must be a path $P_2$ with length 2 in which either $d_{G_1}\left(x_2\right)=1$ or $d_{G_1}\left(x_2\right)=2$. By direct computations, we can easily show that $c\left(G\right)=3$, contrary to $c\left(G\right)=4$. Therefore, this case is also impossible. The proof is complete. □

By Lemma 25, we have

Theorem 11.

Let G be a 4-cutwidth graph with a central cycle $C_6=x_1{x}_2{x}_3{x}_4{x}_5{x}_6{x}_1$. Then G is 4-cutwidth critical if and only if G is one of $\{M_{18},M_{19},M_{20}\}$ in Figure 9, which has a subgraph decomposition $\{{\overline{G}}_1,{\overline{G}}_3,{\overline{G}}_5\}$ of equal cutwidth 3, in which ${\overline{G}}_i={\tau }_2$ or ${\tau }_3$ with central vertex $x_i$ for $i\in \{1,3,5\}$ and there is at least a ${\overline{G}}_{i_0}$ such that ${\overline{G}}_{i_0}={\tau }_2$ with $i_0\in \{1,3,5\}$.

5. 4-Cutwidth Critical Graphs without a Central Vertex and Central Cycle

We now consider the 4-cutwidth critical graphs with neither a central vertex nor a central cycle (see five graphs $M_{21}$–$M_{25}$ in Figure 10).

Theorem 12.

A graph G is 4-cutwidth critical with neither a central vertex nor a central cycle if and only if $G\in \{M_{21},M_{22},M_{23},M_{24},M_{25}\}$.

Proof. 

Sufficiency. For any $G\in \{M_{21},M_{22},M_{23},M_{24},M_{25}\}$, G is needed to show (1) $c\left(G\right)=4$; (2) $c\left(G^{\prime }\right)=3$ for any $G^{\prime }\in \mathcal{M}$$\left(G\right)$. These can be done easily, omitted here. On the other hand, we can see that G has neither the central vertex nor the central cycle.

Necessity. Suppose that G is a 4-cutwidth critical graph without central vertex and central cycle, then G has at least two cycles $C_3$, sharing a common edge. This is because otherwise, G can be thought of as having either a central vertex or a central cycle. So, we have the following:

Claim 2. ${\tau }_5$ in Figure 1 is an edge-induced proper subgraph with cutwidth 3 of G.

By Claim 2, we have

Claim 3. Suppose that H is a 1-connected and minimum 3-cutwidth graph with ${\tau }_5\subset H$, $d_H\left(x_1\right)\le 4$ and $d_H\left(x_3\right)\le 4$, in which $d_H\left(x_i\right)$ is maximum for each $x_i\in V\left({\tau }_5\right)$, then H is graph $\left(a\right)$ in Figure 11.

Claim 4. Suppose that H is a 2-connected and minimum noncritical 3-cutwidth graph with ${\tau }_5\subset H$, then H is graph $\left(b\right)$ in Figure 11.

Figure 11. Two 3-cutwidth graphs containing ${\tau }_5$.

Figure 11. Two 3-cutwidth graphs containing ${\tau }_5$.

By Claims 3 and 4 and the minimality of G, if G is 1-connected, then G must be one of $\{M_{23},M_{24},M_{25}\}$ by direct computations and comparisons. Now, we consider the case that G is 2-connected. Since 4-cutwidth critical graph $M_8$ can be thought of as having a central cycle $C_3$, we can exclude $M_8$ here. Thus, by direct computations and comparisons, G must be one member of $\{M_{21},M_{22}\}$. So, $G\in \{M_{21},M_{22},M_{23},M_{24},M_{25}\}$. □

6. Concluding Remarks

In this paper, we have completely characterized the structural properties of 4-cutwidth critical graphs, from which we can see that except for a handful of irregular critical graphs $M_{21}$–$M_{25}$ in Figure 10, the other 4-cutwidth critical graphs can be classified into two classes: graph class with a central vertex $v_0$, and graph class with a central cycle $C_q$ of length $q\le 6$. By means of some ingenious combination, any member of two classes can achieve a subgraph decomposition $\{H_1,H_2,H_3\}$ (or $\{{\overline{G}}_1,{\overline{G}}_2,{\overline{G}}_3\}$), in which $H_i$ (or ${\overline{G}}_i$) is either a 2-cutwith graph or a 3-cutwidth graph for each $1\le i\le 3$, or a subgraph decomposition $\{H_1,H_2,H_3,H_4\}$ of equal cutwidth 2. For a given integer $k>4$, although it seems difficult to characterize the detailed structures of k-cutwidth critical graphs, some structural properties of some special graph classes can be found. For instance, using [11], any k-cutwidth critical tree with a central vertex $v_0$ has a subtree decomposition $\{T_1,T_2,T_3\}$ of equal cutwidth $k-1$, where, for $1\le i\le 3$, $T_i$ (or $T_i-v_0$) is either a $(k-1)$-cutwidth critical tree or homeomorphic to a $(k-1)$-cutwidth critical tree. Similarly, a k-cutwidth critical non-tree graph $G={\oplus }_{z_0}(G_1,G_2,G_3)$ also has a subgraph decomposition $\{G_1,G_2,G_3\}$ of equal cutwidth $k-1$, and $G_1,G_2,G_3$ are all $(k-1)$-cutwidth critical. In the k-cutwidth critical graphs G with a central cycle $C_q$ of length $q\ge 3$, the structural properties are not yet known. Additionally, for a fixed integer $k_0>4$, finding all the $k_0$-cutwidth critical graph $Gs$ with neither a central vertex nor a central cycle is also a difficult task. All of these are the further objectives to investigate in future works.