On the Relation Between the Domination Number and Edge Domination Number of Trees and Claw-Free Cubic Graphs

Abstract

For a connected graph $G=(V,E)$, the dominating set in graph G is a subset of vertices $F\subset V$ such that every vertex of $V-F$ is adjacent to at least one vertex of F. The minimum cardinality of a dominating set of G, denoted by $\gamma \left(G\right)$, is the domination number of G. The edge dominating set in graph G is a subset of edges $S\subset E$ such that every edge of $E-S$ is adjacent to at least one edge of S. The minimum cardinality of an edge dominating set of G, denoted by ${\gamma }^{\prime }\left(G\right)$, is the edge domination number of G. In this paper, we characterize all trees and claw-free cubic graphs with equal domination and edge domination numbers, respectively.

1. Introduction

In this paper, we only consider simple and undirected graphs without isolated vertices. Let G be a finite simple undirected graph with vertex set V and edge set E. Two edges are adjacent if they are incident to a common vertex, and two vertices u and v are adjacent if there is an edge e incident to both u and v. For $v\in V$, denote by $N_G\left(v\right)$ the open neighborhood of v in G, i.e., $N_G\left(v\right)=\{u\in V|uv\in E\}$ and by $E_G\left(v\right)$ the set of the edges in G incident with v, i.e., $E_G\left(v\right)=\{e\in E|eisincidentwithv\}$. For $e\in E$, denote by $N_G\left(e\right)$ the egde open neighborhood of e in G, i.e., for a subset $X\subseteq V$, we will use the notations $N_G\left(X\right)={\bigcup }_{v\in X}{N}_G\left(v\right)$ and $N_G\left[X\right]=X\cup N_G\left(X\right)$, and the subgraph induced by X is denoted by $G\left[X\right]$. We denote the degree of a vertex v in G by $d_G\left(v\right)$, i.e., $d_G\left(v\right)=\left|N_G\left(v\right)\right|$, and use the notations $\delta \left(G\right)$ and $▵\left(G\right)$ to represent the minimum and maximum degrees of G, respectively. If all the vertices in V have the same degree r, then G is r-regular, or simply regular. A 3-regular graph is called cubic. The distance $d_G(u,v)$ between two vertices u and v in G is the minimum length of a path between u and v, and the distance $d_G(e,e^{\prime })$ between two edges e and $e^{\prime }$ is defined as the smallest distance between the endpoints of e and endpoints of $e^{\prime }$. The diameter, $diam\left(G\right)$, of G is the maximum distance between pairs of vertices in G. A leaf of G is a vertex of degree 1, and its neighbor is called a support vertex, which is not a leaf. A support vertex with at least two leaf neighbors is called a $strongsupportvertex$. A $leafedge$ of G is an edge with one endpoint of degree 1. A rooted treeT distinguishes one vertex r called the root. For each vertex $v\ne r$ of T, a child of v is a neighbor of v away from r.

A path is a nonempty graph $P=(V,E)$ of the form $V=\{x_0,x_1,\dots ,x_k\}$, $E=\{x_0{x}_1,x_1{x}_2,$ $\dots ,x_{k-1}{x}_k\}$, where the $x_i$ are all distinct. The path $P_1$ is the trivial path. A complete graph $K_3$ is called a triangle. The complete graph with four vertices minus one edge is called a diamond. A graph is F-free if it does not contain F as an induced subgraph. In particular, if $F=K_{1,3}$, then the graph is claw-free. If there is no ambiguity in the sequence, the subscript letter in the notation is omitted.

A matching is a subset $M\subseteq E\left(G\right)$ of non-adjacent edges. The matching number is the cardinality of a maximum matching in G, and is denoted by $\mu \left(G\right)$. A subset $X\subseteq V\left(G\right)$ is independent if no edge has both endpoints in X, i.e., $X\cap N\left(X\right)=\varnothing$. The cardinality of a maximum independent set in G, written $\alpha \left(G\right)$, is the independence number of G. The core of G, written core(G), is the intersection of all maximum independent sets in G. A dominating set of G is a subset F of V such that every vertex outside F is adjacent to at least one vertex in F. The domination number $\gamma \left(G\right)$ of G is the minimum cardinality of a dominating set of G. An edge dominating set of G is a subset S of E such that every edge outside S is adjacent to at least one edge in S. The edge domination number ${\gamma }^{\prime }\left(G\right)$ of G is the minimum cardinality of an edge dominating set of G.

There are many relationships between the parameters $\alpha \left(G\right)$ and $\mu \left(G\right)$. It is known that $n-2\mu \left(G\right)⩽n-2{\mu }^{*}\left(G\right)⩽\alpha \left(G\right)⩽n-\mu \left(G\right)$, where ${\mu }^{*}\left(G\right)$ is the number of a minimum maximal matching in G. Boros et al. [1] proved that if G satisfies $\alpha \left(G\right)>\mu \left(G\right)$, then $\alpha \left(G\right)⩽\mu \left(G\right)+\left|core\right(G\left)\right|-1$. Levit et al. [2] then proved that $\alpha \left(G\right)⩽\mu \left(G\right)+\left|core\left(G\right)\right|-|N_G\left(core\left(G\right)\right)|$ where G is a graph with a matching from $N_G\left(core\left(G\right)\right)$ to $core\left(G\right)$. Afterward, Caro et al. [3] proved that $\alpha \left(G\right)⩽\mu \left(G\right)+\left|X\right|-\alpha \left(G\left[N_G\left[X\right]\right]\right)$, where X is any intersection of maximum independent sets in G, and this upper bound is tight. Deniz et al. [4] investigated the maximum independent sets that are disjoint, and proved that if G contains two disjoint maximal independent sets, then $\alpha \left(G\right)⩽\mu \left(G\right)$. Levit et al. [2] also showed that $\alpha \left(G\right)⩽\mu \left(G\right)$ under the condition that G contains a unique odd cycle. Caro et al. [3] extended this and proved that for any G, $\delta \left(G\right)\alpha \left(G\right)⩽\Delta \left(G\right)\mu \left(G\right)$, this bound is tight. At the same time, it was proved that if G is r-regular ($r>0$), then $\alpha \left(G\right)⩽\mu \left(G\right)$. And the following concern is raised: if G is a 3-regular graph, characterize the graph class that satisfies $\alpha \left(G\right)=\mu \left(G\right)$. Mohr et al. [5] characterized the extremal cubic graphs with equal independence number and matching number. Lu et al. [6] characterized r-regular graphs G with $\alpha \left(G\right)=\mu \left(G\right)$ in terms of the Gallai–Edmonds Structure Theorem.

Yannakakis and Gavril [7] proved that the size of the minimum maximal matching is equivalent to the size of the minimum edge dominating set, and Allan et al. [8] showed that if G does not have an induced subgraph isormorphic to $K_{1,3}$, then its domination number is equal to its independent domination number, and the independent domination number is the minimum cardinality of a maximal independent set. Baste et al. [9] conjectured that the domination number $\gamma \left(G\right)$ of a $\delta$-regular graph with $\delta ⩾1$ is at most its edge domination number ${\gamma }^{\prime }\left(G\right)$. They also proved that the conjecture is true for 3-regular claw-free graphs. Civan et al. [10] verified that $\gamma \left(G\right)⩽{\gamma }^{\prime }\left(G\right)$ if G is a claw-free graph with $\delta \left(G\right)⩾2$. And Maniya et al. [11] showed that if G is a fork-free graph with $\delta \left(G\right)⩾2$, then the domination number is at most its edge domination number. Then, this paper considers the characterization of a graph where the domination and edge domination numbers are equal.

The rest of this paper is structured as follows. In the next section, we give some basic definitions and notations that will be used throughout the paper. In Section 2, we give a constructive characterization of trees T with $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$. In Section 3, we characterize all claw-free cubic graphs G with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$. The paper concludes in Section 4 with further work.

Senthilkumar et al. [12] proved that for any tree, its domination number is always no less than its edge domination number.

Theorem 1. 

For any non-trivial tree T, $\gamma \left(T\right)⩾{\gamma }^{\prime }\left(T\right)$.

The following property of connected claw-free graphs is established in [9].

Theorem 2 ([9]). 

$\gamma \left(G\right)⩽{\gamma }^{\prime }\left(G\right)$ if G is a connected claw-free graph with $\delta \left(G\right)⩾2$.

Motivated by the above Theorems 1 and 2, we characterize all trees and claw-free cubic graphs that satisfy the equation in the previous inequalities.

2. Characterization of Trees with Equal Domination Number and Edge Domination Number

In this section, we give a constructive characterization of tree T satisfying $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$. Before we describe the characterization of trees with the same domination and edge domination numbers, we need some preliminary work.

Let G be a graph with a vertex v. An almost edge dominating set of G relative to v is a set of edges of G dominating all edges of G, except $E\left(v\right)$. The almost edge domination number of G relative to v, denoted by ${\gamma }^{\prime }(G;E\left(v\right))$, is the minimum cardinality of an almost edge dominating set of G relative to v.

To characterize the trees with equal domination number and edge domination number, we introduce a family $\mathcal{A}$ of trees $T=T_k$, which can be obtained as follows. Let $T_1$ be a star $S_t$ with $t⩾2$. Let k be a positive integer. Then, $T_{k+1}$ can be obtained recursively from $T_k$ by one of the following operations.

• Operation ${\mathcal{O}}_1$: Add a vertex $u_1$ in $P_3=u_1{u}_2{u}_3$ to any vertex of $T_k$.
• Operation ${\mathcal{O}}_2$: Add a $P_2$ to a vertex v that satisfies the condition ${\gamma }^{\prime }(T;E\left(v\right))={\gamma }^{\prime }\left(T\right)$ or there is no almost edge dominating sets relative to v in $T_k$.
• Operation ${\mathcal{O}}_3$: Add a new vertex to any support vertex of $T_k$.

We now prove that for any tree $T\in \mathcal{A}$, we have $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$.

Observation 1. 

For every connected graph G of diameter at least four, there exists a minimum (respectively, edge) dominating set that contains no leaf vertices (respectively, edges).

Lemma 1. 

If tree $T\in \mathcal{A}$, then $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$.

Proof. 

We use induction on the number k of operations performed to construct the tree T. If $k=0$, i.e., $T_1$ is a star $S_t$ with $t⩾2$, then $\gamma \left(T_1\right)={\gamma }^{\prime }\left(T_1\right)=1$. Assume that the result holds for every tree $T^{\prime }$ of the family $\mathcal{A}$ constructed by $k-1$ operations. Let T be a tree of family $\mathcal{A}$ constructed by k operations. Let F be a minimum dominating set of T without leaf vertices, and S a minimum edge dominating set of T without of leaf edges.

First, assume that T is obtained from $T^{\prime }$ by the operation ${\mathcal{O}}_1$. By the definition of the operation ${\mathcal{O}}_1$, it is easy to know that $\gamma \left(T\right)⩽\gamma \left(T^{\prime }\right)+1$ and ${\gamma }^{\prime }\left(T\right)⩽{\gamma }^{\prime }\left(T^{\prime }\right)+1$. Since the restriction of F (respectively, S) on $T^{\prime }$ is a dominating (respectively, edge dominating) set of $T^{\prime }$, $\gamma \left(T^{\prime }\right)⩽\gamma \left(T\right)-1$ (respectively, ${\gamma }^{\prime }\left(T^{\prime }\right)⩽{\gamma }^{\prime }\left(T\right)-1$). Obviously, $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$.

Second, suppose that T is obtained from $T^{\prime }$ by the operation ${\mathcal{O}}_2$. By the definition of the operation ${\mathcal{O}}_2$, it is easy to know that $\gamma \left(T\right)⩽\gamma \left(T^{\prime }\right)+1$ and ${\gamma }^{\prime }\left(T\right)⩽{\gamma }^{\prime }\left(T^{\prime }\right)+1$.

The restriction of S on $T^{\prime }$ is an edge subset $S_{T^{\prime }}$ such that the edges in $E\left(T^{\prime }\right)-E\left(v\right)$ are dominated or an edge dominating set of $T^{\prime }$. This means $|S_{T^{\prime }}|⩽{\gamma }^{\prime }\left(T\right)-1$ or ${\gamma }^{\prime }\left(T^{\prime }\right)⩽{\gamma }^{\prime }\left(T\right)-1$. Thus ${\gamma }^{\prime }\left(T^{\prime }\right)={\gamma }^{\prime }(T^{\prime };E\left(v\right))⩽\left|S_{T^{\prime }}\right|⩽{\gamma }^{\prime }\left(T\right)-1$, i.e., ${\gamma }^{\prime }\left(T\right)={\gamma }^{\prime }\left(T^{\prime }\right)+1$. Since $\gamma \left(T\right)⩽\gamma \left(T^{\prime }\right)+1={\gamma }^{\prime }\left(T^{\prime }\right)+1={\gamma }^{\prime }\left(T\right)$, by Lemma 1, we have $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$.

Finally, assume that T is obtained from $T^{\prime }$ by the operation ${\mathcal{O}}_3$. By Observation 1, any minimum dominating set of tree $T^{\prime }$ without leaf vertices is a minimum dominating set of tree T, and any minimum edge dominating set of tree $T^{\prime }$ without leaf edges is also a minimum edge dominating set of tree T. Therefore, $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$. □

We now prove that a tree T of order $n⩾2$ satisfying $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$, then it belongs to $\mathcal{A}$.

Lemma 2. 

For any non-trivial tree T with $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$, we have $T\in \mathcal{A}$.

Proof. 

If $diam\left(T\right)⩽2$, T is a star $S_t$, then $T\in \mathcal{A}$. If $diam\left(T\right)=3$, T is a double star whose domination number is 2 but edge domination number is 1, then in the following, we can assume that $diam\left(T\right)⩾4$, that is, the order of the tree T with $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$ is at least 5. We obtain the result by induction on the number of order n. Suppose that every tree $T^{\prime }$ of order $n^{\prime }$ ($5<n^{\prime }⩽n$) with $\gamma \left(T^{\prime }\right)={\gamma }^{\prime }\left(T^{\prime }\right)$ belongs to $\mathcal{A}$.

According to Observation 1, let S (respectively, F) be an edge dominating set (respectively, a dominating set) of T that contains no leaf edges (respectively, no leaf vertices). First, assume that a support vertex of T, say x, is strong. Let y be a leaf adjacent to x. Let $T^{\prime }=T-y$. Since any dominating set (respectively, edge dominating set) containing no leaf vertices (respectively, leaf edges) of $T^{\prime }$ is still a dominating set (respectively, an edge dominating set) of T, we therefore have $\gamma \left(T^{\prime }\right)={\gamma }^{\prime }\left(T^{\prime }\right)$. By the inductive hypothesis, we have $T^{\prime }\in \mathcal{A}$. Then, the tree T can be obtained from $T^{\prime }$ by the operation ${\mathcal{O}}_3$. Thus, $T\in \mathcal{A}$. In the following, we can assume that every support vertex is adjacent to exactly one leaf vertex.

Let $P=v_0{v}_1{v}_2\dots v_t$($t⩾4$) be a diametrical path of T such that $d_T\left(v_2\right)$ is as large as possible. Root T at $v_t$ and consider the child of $v_2$.

Claim 1. 

If $d_T\left(v_2\right)⩾3$, then $v_2$ is not a support vertex.

Suppose that $v_2$ is a support vertex. Since F is a minimum dominating set containing no leaf vertices, $v_1$, $v_2\in F$. Next, we construct a new edge subset $S^{\prime }$ by (1) choosing an incident edge to any vertex in $F-\{v_1,v_2\}$ into $S^{\prime }$ satisfying that for any edge e in $S^{\prime }$, there is always an edge $e^{\prime }$ in $S^{\prime }$ such that $d(e,e^{\prime })⩽2$ and (2) adding the edge $v_1{v}_2$ to $S^{\prime }$. Because F is a dominating set, then $S^{\prime }$ is an edge dominating set of T with cardinality at most $|S^{\prime }|=\gamma \left(T\right)-1$, a contradiction to $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$. Thus, Claim 1 holds.

According to Claim 1, if $v_2$ has a non-leaf child $v_1^{\prime }$ other than $v_1$, we say $v_0^{\prime }$ is the child of $v_1^{\prime }$, and let $T^{\prime }=T-\{v_1^{\prime },v_0^{\prime }\}$. It is obvious that $\gamma \left(T^{\prime }\right)⩽\gamma \left(T\right)-1$ and ${\gamma }^{\prime }\left(T^{\prime }\right)⩽{\gamma }^{\prime }\left(T\right)-1$.

If $t=4$, then there are no almost edge dominating sets relative to $v_2$, and then any minimum edge dominating set of $T^{\prime }$ forms an edge dominating set of T by adding the edges $v_1^{\prime }{v}_2$, so ${\gamma }^{\prime }\left(T\right)⩽{\gamma }^{\prime }\left(T^{\prime }\right)+1$. If $t⩾5$ since $v_2$ lies on one of longest paths P, any almost edge dominating set of $T^{\prime }$ at $v_2$ forms an edge dominating set of T by adding the edges $v_1^{\prime }{v}_2$, so ${\gamma }^{\prime }\left(T\right)⩽{\gamma }^{\prime }(T^{\prime };E\left(v_2\right))+1$. Then, we have ${\gamma }^{\prime }\left(T^{\prime }\right)⩽{\gamma }^{\prime }\left(T\right)-1⩽{\gamma }^{\prime }(T^{\prime };E\left(v_2\right))⩽{\gamma }^{\prime }\left(T^{\prime }\right)$. And it is easy to see that $\gamma \left(T^{\prime }\right)=\gamma \left(T\right)-1$. So we have $\gamma \left(T^{\prime }\right)={\gamma }^{\prime }\left(T^{\prime }\right)$. By the inductive hypothesis, we have $T^{\prime }\in \mathcal{A}$. The tree T can be obtained from $T^{\prime }$ by the operation ${\mathcal{O}}_2$. Thus, $T\in \mathcal{A}$. In the following, we can assume $d_T\left(v_2\right)=2$.

Let $T^{\prime }=T-\{v_2,v_1,v_0\}$. It is easy to know that $\gamma \left(T\right)⩽\gamma \left(T^{\prime }\right)+1$ and ${\gamma }^{\prime }\left(T\right)⩽{\gamma }^{\prime }\left(T^{\prime }\right)+1$. Since the restriction of F on $T^{\prime }$ is also a dominating set of $T^{\prime }$, and the restriction of S containing no edge $v_2{v}_3$ (if S contains edge $v_2{v}_3$, then $S=S-v_2{v}_3+v_3{v}_4$ is a new edge dominating set of T containing no edge $v_2{v}_3$) on $T^{\prime }$ is also an edge dominating set of $T^{\prime }$, that is $\gamma \left(T^{\prime }\right)⩽\gamma \left(T\right)-1$ and ${\gamma }^{\prime }\left(T^{\prime }\right)⩽{\gamma }^{\prime }\left(T\right)-1$. Then, we have $\gamma \left(T^{\prime }\right)={\gamma }^{\prime }\left(T^{\prime }\right)$. By the inductive hypothesis, we have $T^{\prime }\in \mathcal{A}$. The tree T can be obtained from $T^{\prime }$ by operation ${\mathcal{O}}_1$. Thus, $T\in \mathcal{A}$. □

Combining Lemmas 1 and 2, we have the following theorem.

Theorem 3. 

Let T be a non-trivial tree T. Then, $\gamma \left(T\right)={\gamma }^{\prime }\left(T\right)$ if and only if $T\in \mathcal{A}$.

3. Characterization of Claw-Free Cubic Graphs with Equal Domination Number and Edge Domination Number

In this section, we start by providing some additional symbols and some groundwork.

Lemma 3 ([13]). 

Let $G=(V,E)$ be a connected claw-free cubic graph and $G\ne K_4$. Then, vertex set V can be uniquely partitioned into sets such that each induces a triangle or a diamond in G.

According to Lemma 3, the vertex set V of the connected claw-free cubic graph $G\ne K_4$ can be uniquely partitioned into sets, each of which induces a triangle or a diamond in G. Following the notation introduced in [13], such a partition is called a triangle–diamond partition of G, abbreviated as $\Delta$-D-partition. We call each triangle and diamond induced by a set in the $\Delta$-D-partition a unit of the partition. A unit that is a triangle, we call a triangle unit and a unit that is a diamond, we call a diamond unit. (Note that a triangle unit is a triangle that does not belong to a diamond.) We say that two units are adjacent in the $\Delta$-D-partition if there is an edge connecting a vertex of one unit to a vertex of the other unit.

According to the $\Delta$-D-partition of a connected claw-free cubic graph, let $T_{\Delta }$ be any induced triangle in the $\Delta$-D-partition. Any vertex $a\in V\left(T_{\Delta }\right)$, and $a^{\prime }$ is the neighbor of a not in $T_{\Delta }$ and the vertex $a^{″}$ ($a^{″}$ may be equal to $a^{\prime }$) in a triangle $T_{\Delta }^{\prime }$, which is found from the direction of edge $a{a}^{\prime }$, then the induced subgraph of all vertices and edges between a and $a^{″}$ is called a pseudo path, abbreviated as $\overline{P_a}$, as shown in Figure 1. To simplify the representation, a is referred to as the internal endvertex of the pseudo path $\overline{P_a}$ and $a^{\prime \prime }$ as the external endvertex.

Theorem 4. 

Let $G=(V,E)$ be a connected claw-free cubic graph with $G\ne K_4$. Then, $\gamma \left(G\right)⩽t+d$ where t and d are the number of triangle units and diamond units in the Δ-D-partition of G.

Proof. 

Since G is a claw-free cubic graph, by Lemma 3, there is a unique $\Delta$-D-partition of $V\left(G\right)$. Without loss of generality, let t and d be the number of triangle units and diamond units in the $\Delta$-D-partition. Then, we can form a new subset F of E by collecting a vertex in each triangle unit and a vertex of degree 3 in each diamond unit into F. It is easy to see that V is a dominating set of G, i.e., $\gamma \left(G\right)⩽t+d$. □

Lemma 4. 

If the connected claw-free cubic graph G induces a triangle unit T with $V\left(T\right)=\{a,b,c\}$ which has three pseudo paths $\overline{P_a}$, $\overline{P_b}$ and $\overline{P_c}$, then $\gamma \left(G\right)<t+d$, where t and d are the numbers of triangle units and diamond units in the Δ-D-partition of G.

Proof. 

Let T be any triangle unit of G with $V\left(T\right)=\{a,b,c\}$. By definition of the pseudo path, $a^{″}$, $b^{″}$, and $c^{″}$ are the external endvertices of the pseudo paths $\overline{P_a}$, $\overline{P_b}$, and $\overline{P_c}$, respectively. Assume that $a^{″}$, $b^{″}$, and $c^{″}$ belong to the triangles $T^{\prime }$, $T^{″}$, and $T^{‴}$, respectively. Let $a^{\prime }$ be the neighbor of a not in T, $b^{\prime }$ the neighbor of b not in T, and $c^{\prime }$ the neighbor of c not in T.

According to the $\Delta$-D-partition, we can construct a subset F of $V\left(G\right)$ by choosing an arbitrary vertex in each triangle unit and a vertex of degree 3 in each diamond unit. Then, F is a dominating set of G with cardinality $t+d$. Without loss of generality, we assume that $F\cap V\left(T\right)=\left\{a\right\}$. By the construction of F, $|F\cap V(T^{\prime }\left)\right|=1$, $|F\cap V(T^{″}\left)\right|=1$ and $|F\cap V(T^{‴}\left)\right|=1$. Next, we will divide into three cases to discuss whether $a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ belong to the triangle unit or diamond unit.

Case 1. 

$a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ belong to $T_1$, $T_2$, and $T_3$, respectively, as shown in Figure 2.

By the construction of F, $|F\cap V(T\left)\right|=1$, $|F\cap V(T_1\left)\right|=1$, $|F\cap V(T_2\left)\right|=1$, and $|F\cap V(T_3\left)\right|=1$. Let $F_1=\{a^{\prime },b^{\prime },c^{\prime }\}\cup (F-V\left(T\right)-V\left(T_1\right)-V\left(T_2\right)-V\left(T_3\right))$. It is easy to say that $F_1$ is a dominating set of G, i.e., $\gamma \left(G\right)⩽|F_1|<t+d$.

Case 2. 

Some of $a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ belong to the triangle unit, while others belong to the diamond unit as shown in Figure 3.

On one side, without loss of generality, assume that $a^{\prime }$ belongs to a diamond unit, and $b^{\prime }$ and $c^{\prime }$ belong to two different triangle units $T_2$ and $T_3$, illustrated in Figure 3a. By the definition of the pseudo path, we can assume that the diamond unit in pseudo path $\overline{P_a}$ is $D_i$ ($1⩽i⩽k$) with $V\left(D_i\right)=\{a_i,b_i,c_i,d_i\}$, where $a_1=a^{\prime }$ and $a_i{b}_i\notin E$, the edges $b_i{a}_{i+1}$ connecting two adjacent diamond units $D_i$ and $D_{i+1}$.

By the construction of F, $|F\cap D_i|=1$, $|F\cap V(T\left)\right|=1$, $|F\cap V(T_2\left)\right|=1$ and $|F\cap V(T_3\left)\right|=1$. Let $F_2=\{b^{\prime },c^{\prime },a^{″}\}\cup {\bigcup }_{i=1}^k\left\{a_i\right\}\cup (F-V\left(T\right)-V\left(T^{\prime }\right)-V\left(T_2\right)-V\left(T_3\right)-{\bigcup }_{i=1}^{k}V\left(D_i\right))$. It is easy to say that $F_1$ is a dominating set of G, i.e., $\gamma \left(G\right)⩽|F_2|<t+d$.

On the other side, suppose that $a^{\prime }$ belongs to a triangle unit $T_1$, and $b^{\prime }$ and $c^{\prime }$ belong to two different diamond units, illustrated in Figure 3b. Similar to the above, the diamond unit in the pseudo path $\overline{P_b}$ (respectively, $\overline{P_c}$) is $D_i$, where $1⩽i⩽k_b$ (respectively, $1⩽i^{\prime }⩽k_c$) with $V\left(D_i\right)=\{a_{b_i},b_{b_i},c_{b_i},d_{b_i}\}$ (respectively, $V\left(D_{i^{\prime }}\right)=\{a_{c_{i^{\prime }}},b_{c_{i^{\prime }}},c_{c_{i^{\prime }}},d_{c_{i^{\prime }}}\}$), where $a_{b_1}=b^{\prime }$ (respectively, $a_{c_1}=c^{\prime }$) and $a_{b_i}{b}_{b_i}\notin E$ (respectively, $a_{c_{i^{\prime }}}{b}_{c_{i^{\prime }}}\notin E$), the edges $b_{b_i}{a}_{b_{i+1}}$ (respectively, $b_{c_{i^{\prime }}}{a}_{c_{i^{\prime }+1}}$) connecting two adjacent diamond units $D_i$ and $D_{i+1}$ (respectively, $D_{i^{\prime }}$ and $D_{i^{\prime }+1}$).

Let $F_3=\{a^{\prime },b^{″},c^{″}\}\cup {\bigcup }_{i=1}^{k_b}\left\{a_{b_i}\right\}\cup {\bigcup }_{i^{\prime }=1}^{k_c}\left\{a_{c_{i^{\prime }}}\right\}\cup (F-V\left(T\right)-V\left(T_1\right)-V\left(T^{″}\right)-V\left(T^{‴}\right)-{\bigcup }_{i=1}^{k_b}V\left(D_i\right)-{\bigcup }_{i^{\prime }=1}^{k_c}V\left(D_{i^{\prime }}\right))$. Then, $F_3$ is a dominating set of G with cardinality $\left|F\right|-1$, i.e., $\gamma \left(G\right)⩽|F_3|<t+d$.

Case 3. 

$a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ belong to three different diamond units as shown in Figure 4.

Similarly, the diamond unit in the pseudo path $\overline{P_{\theta }}$ is $D_i$, where $1⩽i⩽k_{\theta }$, with $V\left(D_i\right)=\{a_{{\theta }_i},b_{{\theta }_i},c_{{\theta }_i},d_{{\theta }_i}\}$, where $a_{{\theta }_1}={\theta }^{\prime }$ and $a_{{\theta }_i}{b}_{{\theta }_i}\notin E$, and the edges $b_{{\theta }_i}{a}_{{\theta }_{i+1}}$, which connect two adjacent diamond units $D_i$ and $D_{i+1}$, where $\theta \in \{a,b,c\}$.

Let $F_4=\{a^{″},b^{″},c^{″}\}\cup {\bigcup }_{i=1}^{k_a}\left\{a_{a_i}\right\}\cup {\bigcup }_{i^{\prime }=1}^{k_b}\left\{a_{b_{i^{\prime }}}\right\}\cup {\bigcup }_{i^{″}=1}^{k_c}\left\{a_{c_{i^{″}}}\right\}\cup (F-V\left(T\right)-V\left(T^{\prime }\right)-V\left(T^{″}\right)-V\left(T^{‴}\right)-{\bigcup }_{i=1}^{k_a}V\left(D_{a_i}\right)-{\bigcup }_{i^{\prime }=1}^{k_b}V\left(D_{b_{i^{\prime }}}\right)-{\bigcup }_{i^{″}=1}^{k_c}V\left(D_{c_{i^{″}}}\right))$. Then, $F_4$ is a dominating set of G with cardinality $\left|F\right|-1$, that is, $\gamma \left(G\right)⩽|F_4|<t+d$.

To summarize, for every triangle $V\left(T\right)=\{a,b,c\}$ in the $\Delta$-D-partition, if there are three pseudo paths $\overline{P_a}$, $\overline{P_b}$, and $\overline{P_c}$, then $\gamma \left(G\right)<t+d$. □

Combining Theorem 4 and Lemma 4, we have the following corollary.

Corollary 1. 

If G is a connected claw-free cubic graph with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$, then G does not contain such an induced triangle unit T with $V\left(T\right)=\{a,b,c\}$, which has three pseudo paths $\overline{P_a}$, $\overline{P_b}$, and $\overline{P_c}$.

Lemma 5. 

Let G be a connected claw-free cubic graph such that there are two adjacent diamond units in its Δ-D-partition. Then, $\gamma \left(G\right)<{\gamma }^{\prime }\left(G\right)$.

Proof. 

Let d be the number of diamond units in $\Delta$-D-partition. We proceed by induction on d ($d⩾2$). If $d=2$ and $\left|V\right(G\left)\right|=8$, illustrated in Figure 5, it is easy to see that $\gamma \left(G\right)=2$ and ${\gamma }^{\prime }\left(G\right)=3$. In the following, we can assume that $d⩾2$ and $\left|V\right(G\left)\right|>8$.

Suppose that the result is true for all $d=i-1$. For $d=i$, we assume that diamond unit $A_{i-1}$ ($V\left(A_{i-1}\right)=\{a_{i-1},b_{i-1},c_{i-1},d_{i-1}\}$, and $a_{i-1}{b}_{i-1}$ are disadjacent) is adjacent to $A_i$ by connecting the vertex $a_i$ ($\in A_i$) to the vertex $b_{i-1}$ ($\in A_{i-1}$).

Let $G^{\prime }=G-\{a_i,b_i,c_i,d_i\}+b_{i-1}u$, where $u\in N\left(b_i\right)-\{c_i,d_i\}$, $w\in N\left(u\right)-\left\{b_i\right\}$. By the inductive hypothesis, we have $\gamma \left(G^{\prime }\right)<{\gamma }^{\prime }\left(G^{\prime }\right)$.

Obviously, any dominating set of $G^{\prime }$ that adds vertex $c_i$ is a dominating set of G, i.e., $\gamma \left(G\right)⩽\gamma \left(G^{\prime }\right)+1$. Let S be a minimum edge dominating set of G. Then, we have the following claim.

Claim 2. 

$|A_i\cap S|⩽2$.

By contradiction, if $|A_i\cap S|⩾3$, then we can construct a new edge dominating set $S^0$ by deleting edges in $A_i\cap S$, and adding edges $a_i{c}_i$ and $c_i{b}_i$, i.e., $S^0=S-D_i\cap S+a_i{c}_i+c_i{b}_i$, where it is easy to see that $S^0$ is an edge dominating set of G with cardinality of at most $\left|S\right|-1$, a contradiction with the choice of S.

By the above claim, we have divided two cases to be considered, depending on whether $c_i{d}_i\in S$ or not.

Case 1. 

$c_i{d}_i\in S$.

Since $c_i{d}_i\in S$, we have the following claim.

Claim 3. 

$\left|E\right(a_i)\cap S|⩽1$ and $\left|E\right(b_i)\cap S|⩽1$.

By contradiction. Suppose that $\left|E\right(a_i)\cap S|⩾2$, assume $e^{\prime },e^{\prime \prime }\in E\left(a_i\right)\cap S$ and $e^{\prime }\ne b_{i-1}{a}_i$, then $S-e^{\prime }$ is also an edge dominating set of G, a contradiction. So $\left|E\right(a_i)\cap S|⩽1$. Similarly, we have $\left|E\right(b_i)\cap S|⩽1$.

If $|A_i\cap S|=1$, i.e., $A_i\cap S=c_i{d}_i$. If $E\left(a_i\right)\cap S=a_i{b}_{i-1}$, then $S=S-a_i{b}_{i-1}+b_{i-1}{c}_{i-1}$, similarly let $S=S-b_{i}u+uw$ if $E\left(b_i\right)\cap S=b_{i}u$. In a word, we can obtain an edge dominating set S such that $E\left(a_i\right)\cap S=E\left(b_i\right)\cap S=\varnothing$ and $c_i{d}_i\in S$. Then, the restriction of S on $G^{\prime }$ is also an edge dominating set of $G^{\prime }$, so ${\gamma }^{\prime }\left(G^{\prime }\right)⩽{\gamma }^{\prime }\left(G\right)-1$.

If $|A_i\cap S|=2$, combining Claims 2 and 3, then either $\left|\right(E_G\left(a_i\right)-b_{i-1}{a}_i)\cap S|=1$ or $\left|\right(E_G\left(b_i\right)-b_{i}u)\cap S|=1$. Assume $(E_G\left(a_i\right)-b_{i-1}{a}_i)\cap S=\left\{e_0\right\}$, then $S^{\prime }=S-c_i{d}_i-e_0+b_{i-1}u$ is an edge dominating set of $G^{\prime }$, that is ${\gamma }^{\prime }\left(G^{\prime }\right)⩽{\gamma }^{\prime }\left(G\right)-1$. If $\left|\right(E_G\left(b_i\right)-b_{i}u)\cap S|=1$, similarly, we also have ${\gamma }^{\prime }\left(G^{\prime }\right)⩽{\gamma }^{\prime }\left(G\right)-1$.

Case 2. 

$c_i{d}_i\notin S$.

Combining the structure of $A_i$ and Claim 2, then $A_i\cap S\ne \varnothing$, $E_G\left(a_i\right)\cap S\ne \varnothing$ and $E_G\left(b_i\right)\cap S\ne \varnothing$. Let $S^{\prime \prime }$ be the restriction of S on $G^{\prime }$, and $S^{\prime }=S^{\prime \prime }+b_{i-1}u$. Then, $S^{\prime }$ is an edge dominating set of $G^{\prime }$ with cardinality at most $\left|S\right|-2+1$, that is ${\gamma }^{\prime }\left(G^{\prime }\right)⩽{\gamma }^{\prime }\left(G\right)-2+1={\gamma }^{\prime }\left(G\right)-1$.

According to the induction assumption, $\gamma \left(G\right)⩽\gamma \left(G^{\prime }\right)+1<{\gamma }^{\prime }\left(G^{\prime }\right)+1⩽{\gamma }^{\prime }\left(G\right)$, i.e., $\gamma \left(G\right)<{\gamma }^{\prime }\left(G\right)$. □

Combining Theorem 4 and Lemma 5, we have the following corollary.

Corollary 2. 

Let G be a claw-free cubic graph with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$ that contains the diamond unit in its Δ-D-partition. Then, any two diamond units in the Δ-D-partition are disadjacent.

Liu et al. [14] provided the following theorem by carefully constructing counterexamples and analyzing the structural properties of graphs.

Theorem 5 ([14]). 

If G is a connected claw-free cubic graph, then $\alpha \left(G\right)⩽{\displaystyle \frac{1}{3}}\left|V\left(G\right)\right|$. Meanwhile, $\alpha \left(G\right)={\displaystyle \frac{1}{3}}\left|V\left(G\right)\right|$ if and only if $G\in \mathcal{H}$.

Next, we define the family $\mathcal{R}$ of claw-free cubic graph G which can be obtained from connecting the graphs in $\{A,B,C,D\}$ in Figure 6.

For convenience, A (respectively, B, C, and D) is called the A (respectively, B, C, and D)-unit of graph G, where the edge joins a vertex in one unit to a vertex in the other unit by the following ways.

• Operation ${\mathcal{O}}_1$: If a or b (respectively, $e_d$ or $o_d$) is a vertex in A (respectively, D)-unit, suppose a (respectively, $e_d$), then ${\mathcal{O}}_1$ adds the B- or C-unit by an edge join a (respectively, $e_d$) to a vertex e (or o) in the B-unit or a vertex $e_c$ (or $o_c$) in the C-unit.
• Operation ${\mathcal{O}}_2$: At most, one of two endvertices, suppose $e_c$, in chain $P_{C-unit}$ is obtained by replacing the vertices in path P with C-units, then ${\mathcal{O}}_2$ adds the A- or D-unit by edge join $e_c$ to vertex a (or b) in the A-unit or vertex $e_d$ (or $o_d$) in the D-unit.
• Operation ${\mathcal{O}}_3$: If u is an endvertex in chain $P_{AC-unit}$ (or $P_{DC-unit}$) obtained by replacing one vertex in path P with A-unit (or D-unit) and others in P with the C-unit, then ${\mathcal{O}}_3$ adds the B-unit by edge join u to vertex e (or o) in the B-unit.

In the following, we give a graph G obtained by the above operations in Figure 7, and it is easy to see that $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$.

Observation 2. 

Let G be a graph in $\mathcal{R}$.

1.

There are no edges joining a vertex in the A-unit (respectively, D-unit) to a vertex in the A- or D-unit;

2.

At most, one of $e_c$ and $o_c$ in the C-unit is incident to an edge attaching a unit in $\{A,D\}$;

3.

The B-unit can be adjacent to any unit in $\{A,B,C,D\}$;

4.

If there is an edge joining a vertex in the A-unit (respectively, D-unit) to a vertex in the C-unit, then G contains at least one B-unit;

5.

The degree of each vertex in G is 3, and G contains no induced $K_{1,3}$.

In what follows, we define the family $\mathcal{F}$ of claw-free cubic graph $F_1$ and $F_2$ in the Figure 8. By simple validation, $\gamma \left(A\right)={\gamma }^{\prime }\left(A\right)=1$, $\gamma \left(B\right)={\gamma }^{\prime }\left(B\right)=2$, $\gamma \left(C\right)={\gamma }^{\prime }\left(C\right)=3$, $\gamma \left(D\right)={\gamma }^{\prime }\left(D\right)=4$, $\gamma \left(F_1\right)={\gamma }^{\prime }\left(F_1\right)=2$, $\gamma \left(F_2\right)={\gamma }^{\prime }\left(F_2\right)=3$.

For notational convenience, for A-unit $A_i$, $V\left(A_i\right)=\{a_i,b_i,c_i,d_i\}$, where $a_i$ and $b_i$ are disadjacent. Let $T_1$, $T_2$ be the triangles in B-unit $B_i$, and $V\left(T_1\right)=\{e_i,f_i,g_i\}$, $V\left(T_2\right)=\{l_i,h_i,o_i\}$, where $f_i{l}_i$ and $g_i{h}_i$ connect $T_1$ and $T_2$. Let $D_{c_i}$ be the diamond in C-unit $C_i$, $V\left(D_{c_i}\right)=\{a_{c_i},b_{c_i},c_{c_i},d_{c_i}\}$, where $a_{c_i}$ and $b_{c_i}$ are disadjacent. Let $T_1^{\prime }$, $T_2^{\prime }$ be the triangles in $C_i$, and $V\left(T_1^{\prime }\right)=\{e_{c_i},f_{c_i},g_{c_i}\}$, $V\left(T_2^{\prime }\right)=\{l_{c_i},h_{c_i},o_{c_i}\}$, where $f_{c_i}{a}_{c_i}$, $g_{c_i}{h}_{c_i}$, and $b_{c_i}{l}_{c_i}$ connect diamonds and triangles. Let $D_1$ and $D_2$ be two diamonds in D-unit $D_i$, $V\left(D_1\right)=\{a_{d_i},b_{d_i},c_{d_i},d_{d_i}\}$, $V\left(D_2\right)=\{a_{d_i}^{\prime },b_{d_i}^{\prime },c_{d_i}^{\prime },d_{d_i}^{\prime }\}$, where $a_{d_i}$, $b_{d_i}$ and $a_{d_i}^{\prime }$, $b_{d_i}^{\prime }$ are disadjacent, respectively. Let $T_1^{\prime \prime }$, $T_2^{\prime \prime }$ be the triangles in $D_i$, and $V\left(T_1^{\prime \prime }\right)=\{e_{d_i},f_{d_i},g_{d_i}\}$, $V\left(T_2^{\prime \prime }\right)=\{l_{d_i},h_{d_i},o_{d_i}\}$, where $f_{d_i}{a}_{d_i}$, $g_{d_i}{a}_{d_i}^{\prime }$, $b_{d_i}{l}_{d_i}$, and $b_{d_i}^{\prime }{h}_{d_i}$ connect diamonds and triangles, illustrated in Figure 9.

Observation 3. 

The following holds for unit in $\{A,B,C,D\}$.

1.

The edge $c_i{d}_i$ in any A-unit $A_i$ needs at least one edge to dominate, and $\left\{c_i{d}_i\right\}$ is the unique minimum edge dominating set of $A_i$;

2.

The edges in cycle $f_i{g}_i{h}_i{l}_i$ in any B-unit $B_i$ need at least two edges to dominate, and $\{e_i{f}_i,l_i{h}_i\}$, $\{e_i{g}_i,l_i{h}_i\}$, $\{f_i{g}_i,l_i{o}_i\}$, or $\{f_i{g}_i,h_i{o}_i\}$ is a minimum edge dominating set of $B_i$;

3.

The edges $f_{c_i}{g}_{c_i}$, $c_{c_i}{d}_{c_i}$, and $l_{c_i}{h}_{c_i}$ in any C-unit $C_i$ need at least three edges to dominate, and either $\{e_{c_i}{f}_{c_i},c_{c_i}{d}_{c_i}$, $l_{c_i}{h}_{c_i}\}$, or $\{f_{c_i}{g}_{c_i},c_{c_i}{d}_{c_i},l_{c_i}{o}_{c_i}\}$ is the minimum dominating set of $C_i$;

4.

The edges $f_{d_i}{g}_{d_i}$, $c_{d_i}{d}_{d_i}$, $c_{d_i}^{\prime }{d}_{d_i}^{\prime }$, and $l_{d_i}{h}_{d_i}$ in any D-unit $D_i$ need at least four edges to dominate, and $\{f_{d_i}{g}_{d_i},c_{d_i}{d}_{d_i},c_{d_i}^{\prime }{d}_{d_i}^{\prime },l_{d_i}{h}_{d_i}\}$ is the unique minimum edge dominating set of $D_i$.

Next, we show that a graph $G\in \mathcal{R}\cup \mathcal{F}$, then its domination number is equal to its edge domination number.

Lemma 6. 

If G is a claw-free cubic graph in $\mathcal{R}\cup \mathcal{F}$, then $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$.

Proof. 

If $G\in \mathcal{F}$, since $\gamma \left(F_1\right)={\gamma }^{\prime }\left(F_1\right)=2$, $\gamma \left(F_2\right)={\gamma }^{\prime }\left(F_2\right)=3$, it is easy to see that $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$. In what follows, we assume $G\in \mathcal{R}$.

By the structure of the unit in $\{A,B,C,D\}$, we know that (1) vertices $c_i$ and $d_i$ in any A-unit $A_i$ need at least one vertex to dominate; (2) vertices in cycle $f_i{g}_i{h}_i{l}_i$ in any B-unit $B_i$ need at least two vertices to dominate; (3) vertices in three pairwise sharing no common edges $f_{c_i}{g}_{c_i}$, $c_{c_i}{d}_{c_i}$, and $l_{c_i}{h}_{c_i}$ in any C-unit $C_i$ need at least three vertices to dominate; and (4) vertices four pairwise sharing no common edges $f_{d_i}{g}_{d_i}$, $c_{d_i}{d}_{d_i}$, $c_{d_i}^{\prime }{d}_{d_i}^{\prime }$ and $l_{d_i}{h}_{d_i}$ in any D-unit $D_i$ need at least four vertices to dominate. By Observation 2 (5), G is a claw-free cubic graph, then there is a unique $\Delta$-D-partition in G; without loss of generality, assume that t is the number of triangle units and d the number of diamond units in the $\Delta$-D-partition in G. By the construction of G, the triangle unit belongs to the B-unit, C-unit, or D-unit, and the diamond unit belongs to the A-unit, C-unit, or D-unit. Then, $\gamma \left(G\right)⩾t+d$. By Theorem 4, we know $\gamma \left(G\right)=t+d$.

Now, we construct a new subset S of $E\left(G\right)$ by collecting (1) edge $c_i{d}_i$ in each A-unit $A_i$; (2) edges $e_i{f}_i$ and $h_i{o}_i$ in each B-unit $B_i$; (3) edges $e_{c_i}{g}_{c_i}$, $c_{c_i}{d}_{c_i}$, and $l_{c_i}{h}_{c_i}$ in each C-unit $C_i$; and (4) edges $f_{d_i}{g}_{d_i}$, $c_{d_i}{d}_{d_i}$, $c_{d_i}^{\prime }{d}_{d_i}^{\prime }$, and $l_{d_i}{h}_{d_i}$ in each D-unit $D_i$. Then, $\left|S\right|=t+d=\gamma \left(G\right)$. Combining Observations 2 and 3, S is an edge dominating set of G. That is, ${\gamma }^{\prime }\left(G\right)⩽\gamma \left(G\right)$. According to Theorem 2, we know that if $G\in \mathcal{R}$, then $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$. □

In the rest of the paper, we shall prove that if G is a claw-free cubic graph with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$, then it belongs to $\mathcal{R}\cup \mathcal{F}$.

Lemma 7. 

Let G be a claw-free cubic graph with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$. Then, $G\in \mathcal{R}\cup \mathcal{F}$.

Proof. 

Since G is a claw-free cubic graph with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$, then $G\ne K_4$, combining Lemma 3 and Corollary 2, there is a unique $\Delta$-D-partition of G that contains at least one triangle unit. Let t and d be the numbers of triangle units and diamond units in the $\Delta$-D-partition, respectively. By Theorem 4, $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)⩽t+d$. Let T be one of the triangle units in the $\Delta$-D-partition with $V\left(T\right)=\{a,b,c\}$, $a^{\prime }\in N\left(a\right)-T$, $b^{\prime }\in N\left(b\right)-T$, $c^{\prime }\in N\left(c\right)-T$. We proceed further with whether $d=0$ or not.

Case 1. 

$d=0$.

In this case, we know that every unit in the $\Delta$-D-partition is a triangle. Let $\Delta \left(T\right)$ be the set of triangle units in the $\Delta$-D-partition. For $G\ne K_4$ is a claw-free cubic graph without diamond units, by Corollary 1, then $a^{\prime }\ne b^{\prime }$, $b^{\prime }\ne c^{\prime }$, $a^{\prime }\ne c^{\prime }$ and at least two of $a^{\prime }$, $b^{\prime }$ and $c^{\prime }$ belong to the same triangle.

In the following, we divide two subcases to consider whether $a^{\prime }$, $b^{\prime }$ and $c^{\prime }$ belong to the same triangle or not.

Subcase 1.1. 

$a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ belong to the same triangle.

In this case, we know that the edges $a^{\prime }{b}^{\prime }$, $a^{\prime }{c}^{\prime }$, and $b^{\prime }{c}^{\prime }\in G$. Since G is a claw-free cubic graph, it is easy to see that $G=F_1\in \mathcal{F}$, see Figure 10.

Subcase 1.2. 

Exactly two of $a^{\prime }$, $b^{\prime }$ and $c^{\prime }$ belong to the same triangle.

Without loss of generality, we assume that $a^{\prime }$ and $b^{\prime }$ belong to the same triangle $T^{\prime }$ with a vertex $c_{a,b}$ ($\in V\left(T^{\prime }\right)-\{a^{\prime },b^{\prime }\}$). Then, the induced subgraph of $\{a,b,c,a^{\prime },b^{\prime },c_{a,b}\}$ is a B-unit. Since G is a claw-free cubic graph that does not contain diamond units, let $a_{c^{\prime }},b_{c^{\prime }}\in N\left(c^{\prime }\right)-c$, then edge $a_{c^{\prime }}{b}_{c{,}^{\prime }}\in G$. Let $a_{c^{\prime }}^{″}\in N(a_{c^{\prime }}-\{c^{\prime },b_{c^{\prime }}\})$, $b_{c^{\prime }}^{″}\in N(b_{c^{\prime }}-\{c^{\prime },a_{c^{\prime }}\})$. Then, $a_{c^{\prime }}^{″}\ne b_{c^{\prime }}^{″}$, $a_{c^{\prime }}^{″}{b}_{c^{\prime }}^{\prime }\notin E,b_{c^{\prime }}^{″}{a}_{c^{\prime }}^{\prime }\notin E$. Therefore, $a_{c^{\prime }}^{″}$, $b_{c^{\prime }}^{″}$ are adjacent to a common vertex $c^{″}$; otherwise, the induced subgraph $\{a_{c^{\prime }},a_{c^{\prime }}^{″},b_{c^{\prime }}^{″},c^{″}\}$ where $c^{″}\in N\left(a_{c^{\prime }}^{″}\right)-\left\{b_{c^{\prime }}^{″}\right\}$ is a claw. Thus, the induced subgraph of $\{c^{\prime },a_{c^{\prime }},b_{c^{\prime }},a_{c^{\prime }}^{″},b_{c^{\prime }}^{″},c^{″}\}$ is a B-unit. Therefore, we know that G is made up of a series of B-units, i.e., $G\in \mathcal{R}$.

Case 2. 

$d>0$.

By Lemma 5, $t⩾2$. If $t=2$, let $T_1$ and $T_2$ be the triangles in the $\Delta$-D-partition of G. Since G is a claw-free cubic graph, by Lemma 5, the number of diamond units is at most 3, so G is one of the graphs in the following Figure 11.

Since G is a graph with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$, it is easy to see that G is $F_2$, then $G\in \mathcal{F}$.

In the following, we only consider the case where $t⩾3$. Combining $\Delta$-D-partition, Lemmas 4 and 5 and the definitions of A-, B-, C-, D-units, then G can be divided into a series of A-, B-, C-, D-units.

Claim 4. 

Let S be any minimum edge dominating set of G with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$. Then, S contains no edges connecting any two units in $\{A,B,C,D\}$.

By contradiction, if S contains an edge $e^{\prime }$ connecting two units in $\{A,B,C,D\}$, according to Observation 3 (1), (2), (3), (4), regardless of whether the edge connecting two units in $\{A,B,C,D\}$ belong to S or not, $|S\cap A_i|⩾1$, $|S\cap B_i|⩾2$, $|S\cap C_i|⩾3$ and $|S\cap D_i|⩾4$, so ${\gamma }^{\prime }\left(G\right)=\left|S\right|⩾t+d+1$. Since $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)⩽t+d$, it is a contradiction.

Combining Observation 3 (2), (3) and Claim 4, the edges between the B-unit (respectively, C-unit) and the B-unit (or C-unit) can be dominated by some dominating edges in the B-unit (or C-unit). That is, the B-unit (respectively, C-unit) can be adjacent to the B-unit or C-unit, and each each A-unit (respectively, D-unit) is adjacent to a B-unit or C-unit. Therefore, G is constructed by connecting units in $\{B,C\}$ if G does not contain A-unit and D-unit, i.e., $G\in \mathcal{R}$.

Combining Corollary 2 and Claim 4, each A-unit (respectively, D-unit) is adjacent to B-unit or C-unit.

Claim 5. 

If G contains an adjacent A-unit (respectively, D-unit) and C-unit, then G contains at least a B-unit.

By contradiction, suppose G contains no B-units. Combining Observation 3 (1), (3), (4) and Claim 4, then at least one edge connecting two units in G is not dominated, a contradiction to the definition of the edge dominating set.

Claim 6. 

At most, one of two endvertcies of the chain P, created by replacing each vertex with C-unit, appends a unit in $\{A,D\}$.

By contradiction, by Observation 3 (1), (3) and (4), most of the two edges connecting the two endvertcies of the chain P can be dominated by some dominating edges in some C-units. According to Claim 4, the conclusion is valid.

Combining Claims 5 and 6, the graph G with $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$ is in $\mathcal{R}$. □

Combining Lemmas 6 and 7, we have the following theorem:

Theorem 6. 

Let G be a connected claw-free cubic graph. Then, $\gamma \left(G\right)={\gamma }^{\prime }\left(G\right)$ is and only if $G\in \mathcal{R}\cup \mathcal{F}$.

4. Further Work

In Section 4, we characterized all claw-free cubic graphs with equal domination number and edge domination number, there are still many possibilities for further research. Since Mohr et al. [5] characterized the extremal cubic graphs with an equal independence number and matching number, and Lu et al. [6] characterized the r-regular with an equal independence number and matching number, we propose some problems worthy of consideration in the following.

• Characterize claw-free subcubic graphs with an equal domination number and edge domination number;
• Characterize cubic graphs with an equal domination number and edge domination number;
• Characterize graphs with an equal domination number and edge domination number.