Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

Abstract

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and $\left\{b\right\}$-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where $d\ge 2$, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable.

1. Introduction

In the field of computational complexity theory, researchers strive to understand the inherent difficulties of various computational problems and to categorize them based on their solvability and complexity. Three crucial concepts in this domain are polynomial-time solvability, NP-completeness, and parameterized complexity. These concepts help with understanding the efficiency of algorithms.

Polynomial-time solvability refers to problems that can be solved by an algorithm whose running time grows polynomially with the size of the input. This class of problems is denoted by P. Formally, a problem is in P if there exists an algorithm that solves any instance of the problem in time $O\left(n^k\right)$, where n is the size of the input and k is a constant.

NP-completeness is a classification for decision problems that are both in NP (nondeterministic polynomial time) and NP-hard. A problem is in NP if a given solution can be verified in polynomial time. A problem is NP-hard if every problem in NP can be reduced to it in polynomial time.

Parameterized complexity is a sophisticated framework within computational complexity theory that provides a refined analysis of computational problems. Unlike traditional complexity theory, which primarily categorizes problems based on their worst-case time complexity with respect to the total input size, parameterized complexity introduces additional parameters to better understand the structure and computational requirements of problems. This approach allows for more nuanced classification and offers practical insights into solving complex problems efficiently.

A problem instance in parameterized complexity is typically expressed as a pair $(I,k)$, where I is the main input and k is a parameter. A parameter is a specific part of the input that can be isolated and studied independently from the overall input size. For example, in graph problems, parameters might include the number of vertices, the treewidth, or the maximum degree of the graph. A problem is fixed-parameter tractable if it can be solved in time $O(f\left(k\right)·|I{|}^{O\left(1\right)})$, where $f\left(k\right)$ is a computable function that depends only on the parameter k, and $\left|I\right|$ is the size of the input [1]. This means that for small values of k, the problem can be solved efficiently even if the overall input size is large.

An FPT-reduction from a parameterized problem P to another parameterized problem Q is a transformation that converts an instance $(I,k)$ of P into an instance $(I^{\prime },k^{\prime })$ of Q such that:

• The transformation takes time $g\left(k\right)·{\left|I\right|}^{O\left(1\right)}$ for some computable function $g\left(k\right)$;
• The parameter $k^{\prime }$ of the resulting instance $(I^{\prime },k^{\prime })$ is bounded by a computable function $h\left(k\right)$;
• The original instance $(I,k)$ is a yes-instance of P if and only if the transformed instance $(I^{\prime },k^{\prime })$ is a yes-instance of Q.

Para-NP-completeness is a classification for parameterized problems that extends the notion of NP-completeness. A parameterized problem is para-NP-complete if it satisfies the following conditions:

• The problem can be decided by a nondeterministic Turing machine in time $f\left(k\right)·{\left|I\right|}^{O\left(1\right)}$, where $f\left(k\right)$ is a computable function depending only on k, and $\left|I\right|$ is the size of the input;
• The problem remains NP-hard even when the parameter k is fixed.

FPT-reductions are used to demonstrate the para-NP-hardness of problems. If a parameterized problem P can be FPT-reduced to another parameterized problem Q, and P is already known to be para-NP-complete, then Q is also para-NP-hard. This is analogous to classical NP-completeness, where polynomial-time reductions are used to show NP-hardness.

The W-hierarchy is another classification of parameterized problems that extends beyond fixed-parameter tractability. The classes in the W-hierarchy are defined using the concept of weft. The weft is the maximum number of gates with unbounded fan-in on any path from an input to an output gate. A parameterized problem is in the class $W\left[t\right]$ if it is FPT-reducible to the circuit-satisfiability problem with a weft of at most t. If a problem P is in the $W\left[t\right]$ class and every other problem in the same class can be FPT-reduced to P, then P is $W\left[t\right]$-complete.

In this paper, we examine clique transversal problems for d-degenerate graphs with a fixed d, covering polynomial-time solvability, NP-completeness, and parameterized complexity.

A d-degenerate graph is characterized by the property that each induced subgraph of the graph contains a vertex with a degree of at most d. The degeneracy of a graph is the smallest value d such that the graph is d-degenerate [2]. In a d-degenerate graph, every subgraph contains at least one vertex whose degree is d or less, despite the possibility of the graph having vertices with degrees larger than d. The concept of degeneracy provides a way to measure graph sparsity such that d-generate graphs frequently appear in real-world networks where the degree of connectivity of the nodes is not uniformly distributed but follows a pattern that can be encapsulated within the d-degenerate framework.

Degeneracy, or the d-core number [3], has been applied to diverse fields through network modeling and analysis, such as sociology, physics, biology, ecology, and neuroscience. A survey by Malliaros et al. [4] covers its central concepts, essential algorithmic techniques, and the applications mentioned above. Furthermore, d-degenerate graphs are extensively used in theoretical computer science, particularly in graph algorithms and complexity theory. They generalize several well-known classes of graphs, such as planar graphs, regular graphs, graphs of fixed treewidth or degree, and H-minor-free graphs as subclasses [5,6]. This generalization offers a unifying framework to study various properties and algorithms that can be applied to a broad range of graphs.

Over the past two decades, the clique transversal problem and its variations have captured the attention of researchers worldwide. These problems hold theoretical and practical significance, particularly in communication networks [7,8,9,10] and other domains [11,12,13,14,15,16,17]. Despite substantial progress in comprehending the computational complexity of these problems, considerable scope still needs to be explored.

The quest to pinpoint problems solvable in polynomial time continues to drive research, with the aim being to untangle complex computational challenges efficiently. Conversely, categorizing problems as NP-complete sheds light on computational boundaries, prompting the innovation of new algorithmic approaches and paradigms. Exploring the parameterized complexity of these problems through the lens of specific parameters and developing algorithms tailored to utilize these insights promises significant improvements in algorithmic efficiency and practical applicability across diverse fields. This approach refines our understanding of computational complexity and bridges theoretical concepts with real-world problem-solving strategies.

Our paper focuses on critical variations of the clique transversal problem, such as the b-fold clique transversal, the $\left\{b\right\}$-clique transversal, the minus clique transversal, and the signed clique transversal problems. We present a summary of existing complexity results in

Table 1. Previous NP-completeness and polynomial-time results.

Graphs	b-FCTP	$\left\{\mathit{b}\right\}$-CTP	SCTP	MCTP
Chordal	NP-c [11]	NP-c [11]	NP-c [12]	NP-c [13]
Split	NP-c [11]	NP-c [11]	NP-c [11]	NP-c [11]
Planar	−	−	NP-c [13]	NP-c [16]
Suns	P [11]	P [11]	P [11]	P [11]
Balanced	P [11]	P [11]	P [11]	−
$\kappa$-Tree ($\kappa$ is not fixed)	NP-c [11]	−	NP-c [11]	NP-c [11]
Doubly Chordal	NP-c [11]	P [11]	NP-c [12]
Distance Hereditary	P [14] *	P [11]	−	−

* Restricted to the particular case for $b=1$.

and

Table 2. Previous parameterized complexity results.

Parameter: Solution Size or Weight
Graphs	$\mathit{b}$-FCTP	$\left\{\mathit{b}\right\}$-CTP	SCTP	MCTP
Chordal	W[2]-c [16]	−	W[2]-c [16]	W[2]-c [16]
Split	W[2]-c [16]	−	para-NP-c [16]	para-NP-c [16]
Planar	FPT [16]	FPT [16]	para-NP-c [16]	para-NP-c [16]
Doubly Chordal	W[2]-c [16]	−	para-NP-c [16]	−
$\mathcal{G}$-class	W[2]-c [16]	W[2]-c [16]	W[2]-c [16]	W[2]-c [16]
Parameter	$\mathit{b}$-FCTP	$\left\{\mathit{b}\right\}$-CTP	SCTP	MCTP
Treewidth	FPT [16]	FPT [16]	FPT [16]	FPT [16]
	FPT [15] *	FPT [15] *

* Restricted to the particular case for $b=1$.

alongside a comprehensive list of abbreviations and symbols in

Table 3. Abbreviations.

Abbreviation	Meaning
b-FCTP	The b-Folding Clique Transversal Problem
$\left\{b\right\}$-CTP	The $\left\{b\right\}$-Clique Transversal Problem
SCTP	The Signed Clique Transversal Problem
MCTP	The Minus Clique Transversal Problem
NP-c	NP-complete
P	Polynomial-time
W[2]-c	W[2]-complete
para-NP-c	Para-NP-complete
FPT	Fixed-Parameter Tractable
$\mathcal{G}$-class	The class of graphs with a polynomial number of maximal cliques
*	Restricted to the particular case for $b=1$
−	Time complexity remains unknown

.

These problems on d-degenerate graphs present intriguing challenges and opportunities. We aim to comprehensively explore these clique transversal problems for graphs of bounded degeneracy d, navigating through the realms of polynomial-time solvability and the intricacies of parameterized complexity. This paper presupposes a fixed positive integer value for d whenever referring to a graph as d-degenerate. The structure of the remainder of this paper is detailed below:

• Section 2 revisits the problem definitions and fundamental concepts from graph theory relevant to our study.
• Section 3 outlines essential observations that form the foundation for further exploration of the problems associated with d-degenerate graphs.
• Section 4 discusses results achievable in polynomial time for graphs that are 1-degenerate and 2-degenerate.
• Section 5 explores the NP-completeness of these problems in d-degenerate graphs for $d\ge 2$.
• Section 6 examines para-NP-completeness in the context of the signed and minus clique transversal problems within d-degenerate graphs.
• Section 7 establishes the fixed-parameter tractability of the b-fold clique transversal and $\left\{b\right\}$-clique transversal problems in d-degenerate graphs.
• Finally, Section 8 concludes the paper by summarizing our results and proposing directions for future research.

2. Preliminaries

A graph $G=(V,E)$ consists of a set of vertices V and a set of edges E. If the vertices and edges of a graph G are not explicitly specified, $V\left(G\right)$ and $E\left(G\right)$, respectively, are used to refer to them. It is conventional to presume that a graph has n vertices and m edges.

If two vertices are connected by an edge, they are adjacent, or neighbors. The degree of a vertex v in G is the number of neighbors of v in G and is represented by $de{g}_G\left(v\right)$. A vertex v is an isolated vertex in G if $de{g}_G\left(v\right)=0$. We use $N_G\left(v\right)$ to denote the set of neighbors of v in G and use $N_G\left[v\right]$ to denote the set $N_G\left(v\right)\cup \left\{v\right\}$. The minimum and maximum degrees of G, represented by $\delta \left(G\right)$ and $\Delta \left(G\right)$, are $\min \{de{g}_G\left(v\right)\mid v\in V\left(G\right)\}$ and $\max \{de{g}_G\left(v\right)\mid v\in V\left(G\right)\}$, respectively.

If S is a subset of vertices in a graph G, then $G\left[S\right]$ represents the subgraph induced by the vertices in S. For any nonempty subset C of $V\left(G\right)$, the induced subgraph $G\left[C\right]$ is complete if either $\left|C\right|=1$ or any two vertices in $G\left[C\right]$ are adjacent. A subset S of vertices in G is a clique if $G\left[S\right]$ is complete. If S is a clique of b vertices, then S is a b-clique. A clique is maximal if no larger clique contains it. The maximum clique is the clique with the largest number of vertices among all cliques.

The notation $\omega \left(G\right)$ denotes the maximum clique size of a graph, which is the size of the maximum clique in G. The set $C\left(G\right)$ represents the set $\{C\mid C$ and is a maximal clique of $G\}$. A clique transversal set of G is a subset S of $V\left(G\right)$ such that $|S\cap C|\ge 1$ for every maximal clique $C\in C\left(G\right)$. The number ${\tau }_C\left(G\right)$ denotes the clique transversal number of G, which is the minimum size of a clique transversal set of G. The clique transversal problem is to find a minimum clique transversal set of G.

Consider two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. We set $G_1\cup G_2=(V_1\cup V_2,E_1\cup E_2)$. Let $\Phi$ be a set of cliques of a graph G. The clique union graph obtained from $\Phi$ is ${\bigcup }_{C\in \Phi }G\left[C\right]$. If the cliques in $\Phi$ are pairwise disjoint, then it clearly holds that $C\left(H\right)=\Phi$ for the clique union graph H obtained from $\Phi$.

A graph G with six vertices is a $(5,1)$-tadpole graph if the vertices of G can be ordered as $v_1,v_2,\dots ,v_6$ such that $v_1,v_2,\dots ,v_5$ form a cycle and $v_6$ is solely adjacent to $v_1$. The ordering $(v_1,v_2,\dots ,v_6)$ is called a $(5,1)$-tadpole ordering. The $(k,1)$-tadpole graph is also known as a k-pan graph.

A graph G is ℓ-regular if $de{g}_G\left(v\right)=$ ℓ for every $v\in V\left(G\right)$. A 3-regular graph is also called a cubic graph. If $de{g}_G\left(v\right)\le 3$ for every $v\in V\left(G\right)$, then G is a subcubic graph. If there does not exist a 3-clique in G, then G is triangle-free. A graph is considered 3-connected if it has at least four vertices and remains connected whenever any two vertices (and their incident edges) are removed. In simpler terms, you would need to remove at least three vertices to disconnect the graph or to turn it into a trivial graph (a graph with fewer than two vertices).

In a graph G, a subset $D\subseteq V\left(G\right)$ covers an edge $(u,v)$ if D includes at least one u or v. A vertex cover of G is a subset $S\subseteq V\left(G\right)$ such that S covers every edge. The notation $\nu \left(G\right)$ denotes the minimum size of a vertex cover of G. The vertex cover problem is to find a minimum vertex cover of G.

Given a subset Y of real numbers, let $f:X\to Y$ be a function mapping elements from the domain X to the codomain Y. We define $f\left(X^{\prime }\right)$ as ${\sum }_{x\in X^{\prime }}f\left(x\right)$ for any subset $X^{\prime }\subseteq X$. The weight of f is defined as $f\left(X\right)$. From a functional perspective, we can model a clique transversal set of a graph G with a function $f:V\left(G\right)\to \{0,1\}$. The function assigns 1 to vertices in the set and 0 to others. Doing this ensures that $f\left(C\right)\ge 1$ for every maximal clique C in G. This effectively captures the essence of intersecting every maximal clique with at least one vertex.

Definition 1.

Let G be a graph, and let b be a fixed positive integer. A subset S of $V\left(G\right)$ is a b-fold clique transversal set of G if $|S\cap C|\ge b$ for every maximal clique $C\in C\left(G\right)$. The number ${\tau }_C^b\left(G\right)$ represents the minimum size of a b-fold clique transversal set of G. The b-fold clique transversal problem is to find a minimum b-fold clique transversal set of G.

Definition 2.

Let G be a graph, and let b be a fixed positive integer. A function $f:V\left(G\right)\to \{0,1,\dots ,b\}$ is a $\left\{b\right\}$-clique transversal function of G if $f\left(C\right)\ge b$ for every maximal clique $C\in C\left(G\right)$. The number ${\tau }_C^{\left\{b\right\}}\left(G\right)$ denotes the minimum weight of a $\left\{b\right\}$-clique transversal function of G. The $\left\{b\right\}$-clique transversal problem is to find a $\left\{b\right\}$-clique transversal function of G of minimum weight.

Definition 3.

Let G be a graph. A function $f:V\left(G\right)\to \{-1,1\}$ is a signed clique transversal function of G if $f\left(C\right)\ge 1$ for every maximal clique $C\in C\left(G\right)$. The number ${\tau }_C^s\left(G\right)$ represents the minimum weight of a signed clique transversal function of G. The signed clique transversal problem is to find a signed clique transversal function of G of minimum weight.

Definition 4.

Let G be a graph. A function $f:V\left(G\right)\to \{-1,0,1\}$ is a minus clique transversal function of G if $f\left(C\right)\ge 1$ for every maximal clique $C\in C\left(G\right)$. The number ${\tau }_C^{-}\left(G\right)$ denotes the minimum weight of a minus clique transversal function of G. The minus clique transversal problem is to find a minus clique transversal function of G of minimum weight.

3. Some Critical Observations

This section presents some observations as shown in Lemmas 1–8. These critical observations will serve as the basis for the subsequent investigations for the $\left\{b\right\}$-clique transversal, b-fold clique transversal, signed clique transversal, and minus clique transversal problem on d-degenerate graphs.

Let $G=(V,E)$ be a graph with $V=\{v_1,v_2,\dots ,v_n\}$, and let $G_i$ be the subgraph of G induced by $\{v_i,v_{i+1},\dots ,v_n\}$ for every $i\in \{1,2,\cdots ,n\}$. The ordering $(v_1,v_2,\dots ,v_n)$ is a smallest-first ordering of G if $de{g}_{G_i}\left(v_i\right)$ is the minimum degree in $G_i$ for every $i\in \{1,2,\cdots ,n\}$. Such an ordering can be obtained in linear time by using Matula and Beck’s algorithm for constructing a smallest-last ordering of a graph [18].

A graph G is d-degenerate if each induced subgraph of G has a vertex of a degree of at most d. The degeneracy of a graph G is the smallest value d such that G is d-degenerate [2]. Given a smallest-first ordering $(v_1,v_2,\dots ,v_n)$ of a d-degenerate graph G, the degree of $v_i$ in $G_i$ is no more than d for every $i\in \{1,2,\cdots ,n\}$.

Lemma 1.

The maximum clique size of a d-degenerate graph is at most $d+1$.

Proof. 

In a d-degenerate graph, any clique with more than $d+1$ vertices would form an induced subgraph in which each vertex has a degree greater than d. Therefore, the maximum clique size of any d-degenerate graph is at most $d+1$.    □

Lemma 2.

  There exists a d-degenerate graph G such that the graph obtained by removing a vertex from G has the degeneracy d.

Proof. 

Assume that X is a set of vertices $x_1,x_2,\dots ,x_{d+1}$ and that Y is a set of vertices $y_1,y_2,\dots ,y_{d+1}$. We construct a graph G as follows:

(1)

Let $V\left(G\right)=X\cup Y\cup \left\{v\right\}$;

(2)

Let X and Y be two cliques of G;

(3)

Connect $x_i$ to $y_i$ for $2\le i\le d$;

(4)

Connect v to $x_1$ and $y_1$.

Then G is d-degenerate. Figure 1a shows the graph G for $d=2$. Let H be a graph obtained by removing a vertex s from G.

Case 1: Vertex s is the vertex v. Then $G\left[X\right]$ is an induced subgraph of H whose minimum degree is d. Therefore, the degeneracy of H is d.

Case 2: We have either $s\in X$ or $s\in Y$. If $s\in Y$, $G\left[X\right]$ is an induced subgraph of H. Conversely, $G\left[Y\right]$ is an induced subgraph of H if $s\in X$. The minimum degree is d for each subgraph $G\left[X\right]$ and $G\left[Y\right]$. Therefore, the degeneracy of H is d. From the above discussion, the lemma holds.    □

Lemma 3.

There exists a triangle-free, 2-degenerate, subcubic planar graph G such that the graph obtained by removing a vertex from G has the degeneracy 2.

Proof. 

Figure 1b shows an example of a triangle-free, 2-degenerate, subcubic planar graph G. We can prove that the graph obtained by removing a vertex from G has degeneracy of two by an argument similar to that for proving Lemma 2.    □

Lemma 4.

Let ℓ be a fixed positive integer, and let G be a connected ℓ-regular graph with n vertices. If H is a graph obtained by removing a vertex from G, then H is an $(\ell -1)$-degenerate graph.

Proof. 

Based on the definition of an ℓ-regular graph, each vertex in G has the degree ℓ. This indicates that G is ℓ-degenerate. Note that G is connected. No induced subgraph of G with fewer vertices than n is ℓ-regular. Therefore, each proper subgraph of G must have at least one vertex with degree $\ell -1$ or smaller. Hence, the lemma holds.    □

Lemma 5.

  If an n-vertex graph G is triangle-free and contains no isolated vertices, then ${\tau }_C^s\left(G\right)={\tau }_C^2\left(G\right)=n$ and $\nu \left(G\right)={\tau }_C\left(G\right)={\tau }_C^{-}\left(G\right)$.

Proof. 

Since G is triangle-free and contains no isolated vertices, every maximal clique of G is a 2-clique. Consequently, we have ${\tau }_C^2\left(G\right)=n$. Moreover, any signed clique transversal function of G cannot assign $-1$ to any vertex of G. Therefore, ${\tau }_C^s\left(G\right)=n$.

Let f be a minus clique transversal function of G of minimum weight. Based on the previous arguments, it is evident that for each vertex $v\in V\left(G\right)$, $f\left(v\right)$ is either 0 or 1. Therefore, the set S consisting of all the vertices v with $f\left(v\right)=1$ is a clique transversal set of G. We have ${\tau }_C\left(G\right)\le \left|S\right|=f\left(V\left(G\right)\right)={\tau }_C^{-}\left(G\right)$.

On the other hand, let S be a minimum clique transversal set of G. We define a function f such that $f\left(v\right)=1$ for each vertex $v\in S$ and $f\left(v\right)=0$ for each vertex $v\in V\left(G\right)\backslash S$. The function f is a minus clique transversal function. We have ${\tau }_C\left(G\right)=\left|S\right|=f\left(V\left(G\right)\right)\ge {\tau }_C^{-}\left(G\right)$. The above discussion indicates that ${\tau }_C\left(G\right)={\tau }_C^{-}\left(G\right)$.

It is straightforward to verify that a set S of vertices is a minimum vertex cover of G if and only if S also serves as a minimum clique transversal set of G. Then, $\nu \left(G\right)={\tau }_C\left(G\right)$. Hence, the lemma holds.    □

Lemma 6.

  Assume that G is a $(5,1)$-tadpole graph with a $(5,1)$-tadpole ordering $(x_1,x_2,\dots ,x_6)$. Let ℓ and i be fixed positive integers such that $i\in \{1,2,\dots ,\ell \}$. Define ${\tau }_{\ell ,i}^{+}\left(G\right)$ and ${\tau }_{\ell ,i}^{-}\left(G\right)$ so that $\min \left\{f\right(V\left(G\right))\mid f$ is a $\{2\ell \}$-clique transversal function of G with $f\left(x_6\right)=\ell +i\}$ and $\min \left\{f\right(V\left(G\right))\mid f$ is a $\{2\ell \}$-clique transversal function of G with $f\left(x_6\right)=\ell -i\}$, respectively. The following statements hold.

(1) 

${\tau }_C^{\{2\ell \}}\left(G\right)={\tau }_{\ell ,i}^{-}\left(G\right)=6\ell$;

(2) 

${\tau }_{\ell ,i}^{+}\left(G\right)=6\ell +i$.

Proof. 

Let $C_j=\{x_j,x_{j+1}\}$ for $1\le j\le 4$. Let $C_5=\{x_5,x_1\}$ and $C_6=\{x_6,x_1\}$. Clearly, $C\left(G\right)=\{C_j\mid 1\le j\le 6\}$. If f is a $\{2\ell \}$-clique transversal function of G of minimum weight, then $f\left(C_j\right)\ge 2\ell$ for $1\le j\le 6$. We have

$${\tau }_C^{\{2\ell \}}\left(G\right)=\sum _{j=1}^{6}f\left(x_j\right)=f\left(C_2\right)+f\left(C_4\right)+f\left(C_6\right)\ge 6\ell .$$

We define a function h by $h\left(x_j\right)=\ell$ for $1\le j\le 6$. The function h is a $\{2\ell \}$-clique transversal function of G with the weight $6\ell$. Therefore, ${\tau }_C^{\{2\ell \}}\left(G\right)=6\ell .$

Let $f^{\prime }$ be a $\{2\ell \}$-clique transversal function of G such that $f^{\prime }\left(x_6\right)=\ell -i$ and $f^{\prime }\left(V\left(G\right)\right)={\tau }_{\ell ,i}^{-}\left(G\right)$. Since $f^{\prime }$ is a $\{2\ell \}$-clique transversal function of G, we have ${\tau }_{\ell ,i}^{-}\left(G\right)\ge {\tau }_C^{\{2\ell \}}=6\ell$. Let $\widehat{f}$ be a function such that $\widehat{f}\left(x_2\right)=\widehat{f}\left(x_5\right)=\widehat{f}\left(x_6\right)=\ell -i$ and $\widehat{f}\left(x_1\right)=\widehat{f}\left(x_3\right)=\widehat{f}\left(x_4\right)=\ell +i$. The function $\widehat{f}$ is a $\{2\ell \}$-clique transversal function of G with the weight $6\ell$. Therefore, ${\tau }_{\ell ,i}^{-}\left(G\right)=6\ell$. Following the previous discussion on ${\tau }_C^{\{2\ell \}}\left(G\right)$ and ${\tau }_{\ell ,i}^{-}\left(G\right)$, statement (1) holds.

Let $h^{\prime }$ be a $\{2\ell \}$-clique transversal function of G such that $h^{\prime }\left(x_6\right)=\ell +i$ and $h^{\prime }\left(V\left(G\right)\right)={\tau }_{\ell ,i}^{+}\left(G\right)$. Then,

$${\tau }_{\ell ,i}^{+}\left(G\right)=\sum _{j=1}^6{h}^{\prime }\left(x_j\right)=\frac{{\sum }_{j=1}^5{h}^{\prime }\left(C_j\right)}{2}+h^{\prime }\left(x_6\right)\ge 5\ell +(\ell +i)=6\ell +i.$$

Let $\widehat{h}$ be a function such that $\widehat{h}\left(x_6\right)=\ell +i$ and $\widehat{h}\left(x_j\right)=\ell$ for $1\le j\le 5$.

The function $\widehat{h}$ is a $\{2\ell \}$-clique transversal function of G with the weight $6\ell +i$. Therefore, ${\tau }_{\ell ,i}^{+}\left(G\right)=6\ell +i$.    □

Lemma 7.

  Assume that G is a $(5,1)$-tadpole graph with a $(5,1)$-tadpole ordering $(x_1,x_2,\dots ,x_6)$. Let ℓ and i be fixed positive integers such that $i\in \{1,2,\dots ,\ell +1\}$. Define ${\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)$ and ${\widehat{\tau }}_{\ell ,i}^{-}\left(G\right)$ such that $\min \left\{f\right(V\left(G\right))\mid f$ is a $\{2\ell +1\}$-clique transversal function of G with $f\left(x_6\right)=\ell +i\}$ and $\min \left\{f\right(V\left(G\right))\mid f$ is a $\{2\ell +1\}$-clique transversal function of G with $f\left(x_6\right)=\ell -i+1\}$, respectively. The following statements hold.

(1) 

${\tau }_C^{\{2\ell +1\}}\left(G\right)={\widehat{\tau }}_{\ell ,i}^{-}\left(G\right)=6\ell +3$;

(2) 

${\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)=6\ell +3+i$.

Proof. 

Let $C_j=\{x_j,x_{j+1}\}$ for $1\le j\le 4$. Let $C_5=\{x_5,x_1\}$ and $C_6=\{x_6,x_1\}$. Clearly, $C\left(G\right)=\{C_j\mid 1\le j\le 6\}$. If f is a $\{2\ell +1\}$-clique transversal function of G of minimum weight, then $f\left(C_j\right)\ge 2\ell +1$ for $1\le j\le 6$. We have

$${\tau }_C^{\{2\ell +1\}}\left(G\right)=\sum _{j=1}^{6}f\left(x_j\right)=f\left(C_2\right)+f\left(C_4\right)+f\left(C_6\right)\ge 6\ell +3.$$

We define a function h by $h\left(x_1\right)=h\left(x_3\right)=h\left(x_5\right)=\ell +1$ and $h\left(x_2\right)=h\left(x_4\right)=h\left(x_6\right)=\ell$. The function h is a $\{2\ell +1\}$-clique transversal function of G with the weight $6\ell +3$. Therefore, ${\tau }_C^{\{2\ell +1\}}\left(G\right)=6\ell +3.$

Let $f^{\prime }$ be a $\{2\ell +1\}$-clique transversal function of G such that $f^{\prime }\left(x_6\right)=\ell -i+1$ and $f^{\prime }\left(V\left(G\right)\right)={\widehat{\tau }}_{\ell ,i}^{-}\left(G\right)$. Since $f^{\prime }$ is a $\{2\ell +1\}$-clique transversal function of G, we have ${\widehat{\tau }}_{\ell ,i}^{-}\left(G\right)\ge {\tau }_C^{\{2\ell +1\}}=6\ell +3$. Let $\widehat{f}$ be a function such that $\widehat{f}\left(x_2\right)=\widehat{f}\left(x_5\right)=\widehat{f}\left(x_6\right)=\ell -i+1$ and $\widehat{f}\left(x_1\right)=\widehat{f}\left(x_3\right)=\widehat{f}\left(x_4\right)=\ell +i$. The function $\widehat{f}$ is a $\{2\ell +1\}$-clique transversal function of G with the weight $6\ell +3$. Therefore, ${\widehat{\tau }}_{\ell ,i}^{-}\left(G\right)=6\ell +3$. Following the previous discussion on ${\tau }_C^{\{2\ell +1\}}\left(G\right)$ and ${\widehat{\tau }}_{\ell ,i}^{-}\left(G\right)$, statement (1) holds.

Let $h^{\prime }$ be a $\{2\ell +1\}$-clique transversal function of G such that $h^{\prime }\left(x_6\right)=\ell +i$ and $h^{\prime }\left(V\left(G\right)\right)={\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)$. Then,

$${\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)=\sum _{j=1}^6{h}^{\prime }\left(x_j\right)=\frac{{\sum }_{j=1}^5{h}^{\prime }\left(C_j\right)}{2}+h^{\prime }\left(x_6\right)\ge \left(5\ell +\frac{5}{2}\right)+(\ell +i)=6\ell +i+\frac{5}{2}.$$

Since ${\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)$ is a positive integer, ${\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)\ge 6\ell +3+i$. Let $\widehat{h}$ be a function such that $\widehat{h}\left(x_6\right)=\ell +i$, $\widehat{h}\left(x_1\right)=\widehat{h}\left(x_3\right)=\widehat{h}\left(x_5\right)=\ell +1$, and $\widehat{h}\left(x_2\right)=\widehat{h}\left(x_4\right)=\ell$. The function $\widehat{h}$ is a $\{2\ell +1\}$-clique transversal function of G with the weight $6\ell +3+i$. Therefore, ${\widehat{\tau }}_{\ell ,i}^{+}\left(G\right)=6\ell +3+i$.    □

Lemma 8.

  Assume that G is a d-degenerate graph with $C\left(G\right)=\{C_i\mid 1\le i\le p\}$. Let $G_i$ be a $(5,1)$-tadpole graph with a $(5,1)$-tadpole ordering $(x_i^1,x_i^2,\dots ,x_i^6)$ for $1\le i\le p$. Construct H by (1) the union of G and each $G_i$ and (2) connecting $x_i^6$ to all vertices of $C_i$ for $1\le i\le p$. The following statements hold.

(1) 

${\tau }_C^{\{2\ell \}}\left(H\right)={\tau }_C^{\{\ell \}}\left(G\right)+6p\ell$;

(2) 

${\tau }_C^{\{2\ell +1\}}\left(H\right)={\tau }_C^{\{\ell +1\}}\left(G\right)+(6\ell +3)·p.$

Proof. 

(1) Let ${\widehat{C}}_i=C_i\cup \left\{x_i^6\right\}$ for $1\le i\le p$. By the construction of H, we have $C\left(H\right)=\left({\bigcup }_{i=1}^{p}C\left(G_i\right)\right)\cup \{{\widehat{C}}_i\mid 1\le i\le p\}$. Assume that f is an $\{\ell \}$-clique transversal function of G of minimum weight. Let h be a function of H such that $h\left(v\right)=f\left(v\right)$ for every $v\in V\left(G\right)$, and $h\left(x_i^j\right)=\ell$ for $1\le j\le 6$ and $1\le i\le p$. The function h is a $\{2\ell \}$-clique transversal function of H. We have ${\tau }_C^{\{2\ell \}}\left(H\right)\le {\tau }_C^{\{\ell \}}\left(G\right)+6p\ell .$

Conversely, we assume that h is a $\{2\ell \}$-clique transversal function of H of minimum weight. We consider $h\left(x_j^6\right)$ for $1\le j\le p$.

Case 1: $h\left(x_j^6\right)=\ell$. Since $h\left({\widehat{C}}_j\right)=h\left(x_j^6\right)+h\left(C_j\right)\ge 2\ell$, we have $h\left(C_j\right)\ge \ell$. Moreover, $h\left(V\left(G_j\right)\right)=6\ell$ by the weight minimality of h and statement (1) in Lemma 6.

Case 2: $h\left(x_j^6\right)<\ell$. Assume that $h\left(x_j^6\right)=\ell -i$, where $i\in \{1,2,\dots ,\ell \}$. Since $h\left({\widehat{C}}_j\right)=h\left(x_j^6\right)+h\left(C_j\right)\ge 2\ell$, we have $h\left(C_j\right)\ge \ell +i$. Moreover, $h\left(V\left(G_j\right)\right)=6\ell$ by the weight minimality of h and statement (1) in Lemma 6.

Case 3: $h\left(x_j^6\right)>\ell$. Assume that $h\left(x_j^6\right)=\ell +i$, where $i\in \{1,2,\dots ,\ell \}$. Since $h\left({\widehat{C}}_j\right)=h\left(x_j^6\right)+h\left(C_j\right)\ge 2\ell$, we have $h\left(C_j\right)\ge \ell -i$. Moreover, $h\left(V\left(G_j\right)\right)=6\ell +i$ by the weight minimality of h and statement (2) in Lemma 6. In this case, $h\left(C_j\right)$ must be no more than ℓ. Otherwise, the function $h^{\prime }$, defined by $h^{\prime }\left(x_j^k\right)=\ell$ for $1\le k\le 6$ and $h^{\prime }\left(v\right)=h\left(v\right)$ for every $v\in V\left(H\right)\backslash V\left(G_j\right)$, would be a $\{2\ell \}$-clique transversal function of H with smaller weight than the weight of h. Therefore, there exists a $\{2\ell \}$-clique transversal function $\widehat{h}$ of H satisfying the following condition:

(i)

$\widehat{h}\left(x_j^6\right)=h\left(x_j^6\right)-i=\ell$, and $\widehat{h}\left(x_j^k\right)=\ell$ for $1\le k\le 5$;

(ii)

$\widehat{h}\left(v\right)\ge h\left(v\right)$ for every $v\in C_j$, and $\widehat{h}\left(C_j\right)=h\left(C_j\right)+i\ge \ell$;

(iii)

$\widehat{h}\left(v\right)=h\left(v\right)$ for every $v\in V\left(H\right)\backslash \left(V\left(G_j\right)\cup C_j\right)$;

(iv)

$\widehat{h}\left(V\left(H\right)\right)=h\left(V\left(H\right)\right)$.

From the discussion of the three cases, we conclude that there exists a $\{2\ell \}$-clique transversal function $\widehat{h}$ of H such that $\widehat{h}\left(V\left(G_j\right)\right)=6\ell$ and $\widehat{h}\left(C_j\right)\ge \ell$ for $1\le j\le p$. Let $\widehat{f}$ be the function of G such $\widehat{f}\left(v\right)=\widehat{h}\left(v\right)$ for every $v\in V\left(G\right)$. Then, the function $\widehat{f}$ is an $\{\ell \}$-clique transversal function of G. We have

$${\tau }_C^{\{2\ell \}}\left(G\right)=\widehat{h}\left(V\left(G\right)\right)+\sum _{i=1}^p\widehat{h}\left(V\left(G_i\right)\right)\ge \widehat{f}\left(V\left(G\right)\right)+6p\ell \ge {\tau }_C^{\left\{d\right\}}\left(G\right)+6p\ell .$$

Following what we have discussed above, we obtain that ${\tau }_C^{\{2\ell \}}\left(H\right)={\tau }_C^{\{\ell \}}\left(G\right)+6p\ell .$ Statement (1) therefore holds.

(2) Statement (2) can be verified using Lemma 7 and the same proof technique as statement (1) in this lemma.    □

4. Polynomial-Time Results

This section presents polynomial-time results for 1-degenerate and 2-degenerate graphs.

Theorem 1.

The following statements hold for any 1-degenerate graph G of n vertices without isolated vertices.

(1) 

${\tau }_C^2\left(G\right)={\tau }_C^s\left(G\right)=n$;

(2) 

${\tau }_C^{\left\{b\right\}}\left(G\right)=b·{\tau }_C\left(G\right)$;

(3) 

${\tau }_C\left(G\right)={\tau }_C^{-}\left(G\right)$;

(4) 

${\tau }_C\left(G\right)$, ${\tau }_C^{-}\left(G\right)$, and ${\tau }_C^{\left\{b\right\}}\left(G\right)$ are linear-time solvable.

Proof. 

A forest is a graph whose connected component is a tree. Based on the definition of a 1-degenerate graph, G is a forest, and thus, it is triangle-free. By Lemma 5, we have ${\tau }_C^2\left(G\right)={\tau }_C^s\left(G\right)=n$ and ${\tau }_C\left(G\right)={\tau }_C^{-}\left(G\right)$. Since forests are distance-hereditary, ${\tau }_C\left(G\right)$ can be computed in linear time [14]. Therefore, ${\tau }_C\left(G\right)$ and ${\tau }_C^{-}\left(G\right)$ are linear-time solvable for any 1-degenerate graph G.

Lee shows that distance-hereditary graphs are clique perfect [14] and that ${\tau }_C^{\left\{b\right\}}\left(H\right)=b·{\tau }_C\left(H\right)$ for any clique perfect graph H [11]. Therefore, ${\tau }_C^{\left\{b\right\}}\left(G\right)$ is equal to $b·{\tau }_C\left(G\right)$ and is linear-time solvable.    □

Theorem 2.

The signed clique transversal problem is linear-time solvable for 2-degenerate graphs.

Proof. 

Consider a 2-degenerate graph G with n vertices. It has $O\left(n\right)$ maximal cliques, which can be listed in linear time [19]. Lemma 1 indicates that the largest clique in G does not exceed three vertices. We categorize the maximal cliques of G into two sets $S_1$ and $S_2$, where $S_1=\{C\mid C\in C\left(G\right),|C|\le 2\}$ and $S_2=\{C\mid C\in C\left(G\right),|C|=3\}$. We can obtain these sets in $O\left(n\right)$ time.

Define $V_i$ as a set of vertices formed by the union of all maximal cliques in $S_i$ for $i=1,2$. Consequently, $V\left(G\right)$ is the union of $V_1$ and $V_2$. We then construct the clique union graph H obtained from S. It is evident that $C\left(H\right)=S_2$ and H is 2-degenerate. Let $(v_{b_1},v_{b_2},\dots ,v_{b_p})$ be a smallest-first ordering of H. By applying Matula and Beck’s algorithm [18], we generate a smallest-last ordering $(v_{b_1},v_{b_2},\dots ,v_{b_p})$ of H in linear time.

Since every maximal clique in $S_1$ contains no more than two vertices, a signed clique transversal function of G does not assign the value $-1$ to any vertex in $V_1$. Hence, we give a value of 1 to all vertices of $V_1$ in $O\left(n\right)$ time. Next, we detail the process for assigning a value from the set $\{1,-1\}$ to each vertex in $V_2$.

We visit each vertex of $V_2$ following the smallest-first ordering $(v_{b_1},v_{b_2},\dots ,v_{b_p})$ of H. Note that $V_1$ and $V_2$ are not necessarily mutually exclusive. If a vertex $v_{b_j}$ already has the assigned value of 1, we proceed to visit the next vertex. Otherwise, we assign $-1$ to $v_{b_j}$ and assign 1 to each unvalued neighboring vertex $v_{b_{\ell }}$ with $\ell >j$. Due to H’s 2-degeneracy, $v_{b_j}$ is adjacent to at most two vertices with larger indices. The assignment can be completed in $O\left(n\right)$ time.

Let $f:V\left(G\right)\to \{-1,1\}$ be the function of G fulfilling the aforementioned assignment. One can easily verify that f is a signed clique transversal function of G and can be accomplished in linear time. The following shows that f is a signed clique transversal function of G of minimum weight.

Let h be a signed clique transversal function of G of minimum weight such that $\left|\right\{v\in V\left(G\right)\mid f\left(v\right)=h\left(v\right)\left\}\right|$ is minimum. Clearly, $f\left(v\right)=h\left(v\right)$ for every $v\in V_1$. We next consider each vertex $v_{b_i}\in V_2$ for $i=1,2,\dots ,p$.

Assume that $\left|\right\{v\in V\left(G\right)\mid f\left(v\right)=h\left(v\right)\left\}\right|\ne 0$. There exists an index j such that $f\left(v_{b_j}\right)\ne h\left(v_{b_j}\right)$ and $f\left(v_{b_i}\right)=h\left(v_{b_i}\right)$ if $i<j$. Since $f\left(v_{b_j}\right)\ne h\left(v_{b_j}\right)$, $v_{b_j}$ is in $V_2\backslash V_1$.

Case 1: $f\left(v_{b_j}\right)=1$ and $h\left(v_{b_j}\right)=-1$. Recall that $v_{b_j}\in V_2\backslash V_1$. Following our assignment, we allocate the value 1 to $v_{b_j}$ because there exists an index $i<j$ such that $v_{b_i}$ is a neighbor of $v_{b_j}$, with $f\left(v_{b_i}\right)=-1$. Since $i<j$, $h\left(v_{b_i}\right)=f\left(v_{b_i}\right)=-1$. Let C be a maximal clique of $S_2$ containing $v_{b_i}$ and $v_{b_j}$. Note that every maximal clique of $S_2$ consists of three vertices. Then, $h\left(C\right)<1$, which contradicts that h is a signed clique transversal function of G. Thus, $h\left(v_{b_j}\right)=f\left(v_{b_j}\right)=1$.

Case 2: $f\left(v_{b_j}\right)=-1$ and $h\left(v_{b_j}\right)=1$. Since $v_{b_j}\in V_2\backslash V_1$, we assign $-1$ to $v_{b_j}$ following our assignment, as it had not been given any value when visited. Additionally, there is no neighboring vertex $v_{b_i}$ of $v_{b_j}$, with $f\left(v_{b_i}\right)=-1$, for $i<j$.

Let $h^{\prime }$ be the function defined by $h^{\prime }\left(v_{b_j}\right)=-1$ and $h^{\prime }\left(v\right)=h\left(v\right)$ for every $v\in V\left(G\right)\backslash \left\{v_{b_j}\right\}$. If $h\left(C\right)=3$ for every maximal clique C in $S_2$ containing $v_{b_j}$, then $h^{\prime }\left(C\right)=1$, and $h^{\prime }$ is a signed clique transversal function of G with $h^{\prime }\left(V\left(G\right)\right)<h\left(V\left(G\right)\right)$. It contradicts that h is a signed clique transversal function of G of minimum weight. Hence, there must be a maximal clique C in $S_2$ containing $v_{b_j}\in C$ with $h\left(C\right)=1$. Let $C=\{v_{b_j},v_{b_k},v_{b_{\ell }}\}$ such that $h\left(v_{b_{\ell }}\right)=-1$ and $h\left(v_{b_k}\right)=1$. Note that $f\left(v_{b_j}\right)=-1$. Then, $f\left(v_{b_{\ell }}\right)=1$ and $f\left(v_{b_k}\right)=1$. With all indices $i<j$ satisfying $h\left(v_{b_i}\right)=f\left(v_{b_i}\right)$, it follows that $j<\ell$. By the smallest-first ordering, $v_{b_j}$ has at most two neighbors with larger indices.

Define $\widehat{h}$ as the function for which $\widehat{h}\left(v_{b_j}\right)=-1$, $\widehat{h}\left(v_{b_{\ell }}\right)=1$, and $\widehat{h}\left(v\right)=h\left(v\right)$ for every $v\in V\left(G\right)\backslash \{v_{b_j},v_{b_{\ell }}\}$. Clearly, the function $\widehat{h}$ is a signed clique transversal function and maintains the same weight as h and $|\{v\in V\left(G\right)\mid f\left(v\right)=\widehat{h}\left(v\right)\}| < |\{v\in V\left(G\right)\mid f\left(v\right)=h\left(v\right)\}|$. It contradicts the assumption that h is a signed clique transversal function of G of minimum weight such that $\left|\right\{v\in V\left(G\right)\mid f\left(v\right)=h\left(v\right)\left\}\right|$ is minimum.

From the above discussion, the function f is a signed clique transversal function of G of minimum weight.    □

5. NP-Completeness Results

This section presents NP-completeness results for d-degenerate graphs with $d\ge 2$.

Theorem 3.

  The clique transversal and the minus clique transversal problems are NP-complete when restricted to triangle-free, 3-connected, cubic planar graphs.

Proof. 

Let G be a triangle-free, 3-connected, cubic planar graph. Lemma 5 indicates that $\nu \left(G\right)={\tau }_C\left(G\right)={\tau }_C^{-}\left(G\right)$. The vertex cover problem is NP-complete when restricted to triangle-free, 3-connected, cubic planar graphs [20]. Hence, the theorem holds.    □

Theorem 4.

The clique transversal and the minus clique transversal problems are NP-complete when restricted to triangle-free, 2-degenerate, subcubic planar graphs.

Proof. 

Let G be a triangle-free, 2-degenerate, subcubic planar graph. Let $f:V\left(G\right)\to \{-1,0,1\}$ be a function of G. By Lemma 1, the maximum clique size of G is no more than 3. Eppstein et al. [19] demonstrated that a 2-degenerate graph has $O\left(n\right)$ maximal cliques, each of which can be enumerated in linear time. Consequently, verifying whether a function f is a minus clique transversal function of G takes linear time by confirming that $f\left(C\right)$ is at least 1 for every maximal clique C in G. Furthermore, determining if the weight of f is no more than a specific integer k is achieved in $O\left(n\right)$ time by calculating the sum of the weights of all vertices and then comparing the sum to k. Hence, the minus clique transversal problem on triangle-free, 2-degenerate, subcubic planar graphs is in NP. Similarly, the argument above can be applied to prove that the clique transversal problem on the same type of graphs is in NP.

Theorem 3 shows that the clique transversal problem is NP-complete when restricted to triangle-free, 3-connected, cubic planar graphs. Subsequently, we reduce the clique transversal problem on triangle-free, 3-connected, cubic planar graphs to the clique transversal problem on triangle-free, 2-degenerate, planar graphs.

Let G be a triangle-free, 3-connected, cubic planar graph. Then, G has the following properties.

(1)

The maximum clique size of G is 2;

(2)

After removing any two vertices of G, the resulting graph remains connected;

(3)

The degree of every vertex of G is 3.

Let H be the graph obtained by removing a vertex v from G. The graph H is planar and subcubic. Moreover, the graph H is connected and 2-degenerate according to the 3-connectivity of G and Lemma 4. Therefore, H is a triangle-free, 2-degenerate, subcubic planar graph.

Assume that S is a minimum clique transversal set of G. Let $v_1$, $v_2$, and $v_3$ be the neighbors of v in G. We consider the following cases.

Case 1: $v\in S$. Note that H is connected and $C\left(H\right)=C\left(G\right)\backslash \{\{v,v_i\}\mid 1\le i\le 3\}$. Therefore, ${\tau }_C\left(G\right)={\tau }_C\left(H\right)+1$.

Case 2: $v\notin S$. Then, S contains $v_1$, $v_2$, and $v_3$. Let $H^{\prime }$ be the graph obtained by removing $v_1,v_2,v_3$ from H. If $H^{\prime }$ contains an isolated vertex $v^{\prime }$, then $v^{\prime }$ is adjacent to all the vertices $v_1,v_2,v_3$ in G. Clearly, $v^{\prime }\notin S$ by the size minimality of S. Let $\widehat{H}$ be the graph obtained by removing every isolated vertex from $H^{\prime }$. Therefore, ${\tau }_C\left(G\right)={\tau }_C\left(\widehat{H}\right)+3$. If $H^{\prime }$ does not contain an isolated vertex, then ${\tau }_C\left(G\right)={\tau }_C\left(H^{\prime }\right)+3$.

Following the discussion above, we have

$${\tau }_C\left(G\right)=\left\{\begin{array}{c}\min \{{\tau }_C\left(H\right)+1,{\tau }_C\left(\widehat{H}\right)+3\}\mathrm{if} H^{\prime } \mathrm{contains} \mathrm{an} \mathrm{isolated} \mathrm{vertex},\hfill \\ \min \{{\tau }_C\left(H\right)+1,{\tau }_C\left(H^{\prime }\right)+3\}\mathrm{otherwise}.\hfill \end{array}\right.$$

The reduction runs in polynomial time and shows that we can use the solutions to the clique transversal problem for triangle-free, 2-degenerate, subcubic graphs to solve the clique transversal problem for triangle-free, 3-connected, cubic planar graphs. Hence, the clique transversal problem on triangle-free, 2-degenerate, subcubic planar graphs is NP-complete.

Consider the aforementioned graphs H, $H^{\prime }$, and $\widehat{H}$. They are all triangle-free and 2-degenerate. By Lemma 5, we know that ${\tau }_C^{-}\left(G\right)={\tau }_C\left(G\right)$, ${\tau }_C^{-}\left(H\right)={\tau }_C\left(H\right)$, ${\tau }_C^{-}\left(H^{\prime }\right)={\tau }_C\left(H^{\prime }\right)$, and ${\tau }_C^{-}\left(\widehat{H}\right)={\tau }_C\left(\widehat{H}\right)$. We have

$${\tau }_C^{-}\left(G\right)=\left\{\begin{array}{c}\min \{{\tau }_C^{-}\left(H\right)+1,{\tau }_C^{-}\left(\widehat{H}\right)+3\}\mathrm{if} H^{\prime } \mathrm{contains} \mathrm{an} \mathrm{isolated} \mathrm{vertex},\hfill \\ \min \{{\tau }_C^{-}\left(H\right)+1,{\tau }_C^{-}\left(H^{\prime }\right)+3\}\mathrm{otherwise}.\hfill \end{array}\right.$$

Hence, the minus clique transversal problem on triangle-free, 2-degenerate, subcubic planar graphs is NP-complete.    □

Theorem 5.

The signed clique transversal problem on 3-degenerate graphs is NP-complete.

Proof. 

We can prove that the signed clique transversal problem on 3-degenerate graphs is in NP by an argument similar to that of the proof in Theorem 4.

Theorem 4 shows that the clique transversal problem is NP-complete for triangle-free, 2-degenerate, subcubic planar graphs. Subsequently, we reduce the clique transversal problem on triangle-free, 2-degenerate, subcubic planar graphs to the signed clique transversal problem on 3-degenerate graphs.

Let $G=(V,E)$ be a triangle-free, 2-degenerate, subcubic planar graph with n vertices. Every maximal clique in $C\left(G\right)$ is a 2-clique. We construct a graph $\widehat{G}$ with $V\left(\widehat{G}\right)=V\cup \{x,y\}$ and $E\left(\widehat{G}\right)=E\left(G\right)\cup \{(x,v)\mid v\in V\cup \left\{y\right\}\}$. Then, $\widehat{G}$ is 3-degenerate, and $C\left(\widehat{G}\right)=\left\{\{x,y\}\right\}\cup \{\{u,v,x\}\mid (u,v)\in E\left(G\right)\}$.

Let S be a minimum clique transversal set of G. Define f as the function of $\widehat{G}$, where $f\left(v\right)=1$ for all vertices $v\in S\cup \{x,y\}$, and $f\left(v\right)=-1$ for the other vertices v in $\widehat{G}$. Then, f is a signed clique transversal function of H. We have ${\tau }_C^s\left(\widehat{G}\right)\le 2{\tau }_C\left(G\right)+2-n$.

Conversely, we consider a signed clique transversal function h of $\widehat{G}$ of minimum weight. Evidently, $h\left(x\right)=h\left(y\right)=1$. Except for $\{x,y\}$, each maximum in $C\left(\widehat{G}\right)$ consists of x and two vertices $u,v$ in G with $(u,v)\in E\left(G\right)$. For each vertex $(u,v)\in E\left(G\right)$, h assigns $-1$ to at most one vertex of u and v. Therefore, the set $D=\{v\in V(G)\mid h(v)=1\}$ is a clique transversal set of G. We have ${\tau }_C\left(G\right)\le \left|D\right|=({\tau }_C^s\left(\widehat{G}\right)+n-2)/2$. Following the discussion above, we have ${\tau }_C\left(G\right)=({\tau }_C^s\left(\widehat{G}\right)+n-2)/2$.

The reduction runs in polynomial time and shows that ${\tau }_C\left(G\right)\le k$ if and only if ${\tau }_C^s\left(\widehat{G}\right)\le 2k+2-n$. Hence, the signed clique transversal problem on 3-degenerate graphs is NP-complete.    □

Corollary 1.

  Assume that d is a fixed integer larger than 2. The clique transversal and the minus clique transversal problems are NP-complete for d-degenerate graphs.

Proof. 

A $d_1$-degenerate graph is d-degenerate if $d_1<d$. By Theorem 4, the corollary holds.    □

Corollary 2.

  Assume that d is a fixed integer larger than 3. The signed clique transversal problem is NP-complete for d-degenerate graphs.

Proof. 

A $d_1$-degenerate graph is d-degenerate if $d_1<d$. By Theorem 5, the corollary holds.    □

Corollary 3.

Let b and d be two fixed integers such that $b>1$ and $d>2$. The b-fold clique transversal problem is NP-complete for d-degenerate graphs.

Proof. 

We assume that G is a $(d-1)$-degenerate graph. We construct a graph H by adding a vertex x to G and connecting x to all vertices in G. Then, H is a d-degenerate graph.

Consider a minimum b-fold clique transversal set S of H. Let $v\in S$. If S does not contain x, the set $S^{\prime }=(S\backslash \left\{v\right\})\cup \left\{x\right\}$ is also a minimum b-fold clique transversal set of H. We therefore assume that S contains x. Then, the set $S\backslash \left\{x\right\}$ is a $(b-1)$-fold clique transversal set of G. We have ${\tau }_C^{b-1}\left(G\right)\le {\tau }_C^b\left(H\right)-1$. Conversely, we consider a minimum $(b-1)$-fold clique transversal set D of G. The set $D\cup \left\{x\right\}$ is a b-fold clique transversal set of H. We have ${\tau }_C^b\left(H\right)\le {\tau }_C^{b-1}\left(G\right)+1$.

Following what we discussed above, we have ${\tau }_C^b\left(H\right)={\tau }_C^{b-1}\left(G\right)+1$. Therefore, ${\tau }_C^b\left(H\right)\le k$ if and only if ${\tau }_C^{b-1}\left(G\right)\le k-1$. Note that $b>1$ and $d>2$. Since ${\tau }_C^1\left(G\right)={\tau }_C\left(G\right)$ and the clique transversal problem is NP-complete for 2-degenerate graphs, as shown in Theorem 4, the b-fold clique transversal problem is NP-complete for d-degenerate graphs.    □

Theorem 6.

The $\left\{2\right\}$-clique transversal problem is NP-complete for 2-degenerate graphs.

Proof. 

We can prove that the $\left\{2\right\}$-clique transversal problem on 2-degenerate graphs is in NP by an argument similar to that of the proof in Theorem 4.

Let G be a triangle-free, 2-degenerate, subcubic planar graph with $C\left(G\right)=\{C_i\mid 1\le i\le p\}$. Let $G_i$ be a $(5,1)$-tadpole graph with a $(5,1)$-tadpole ordering $(x_i^1,x_i^2,\dots ,x_i^6)$ for $1\le i\le p$. We construct H by (1) the union of G and each $G_i$ and (2) connecting $x_i^6$ to all vertices of $C_i$ for $1\le i\le p$. Clearly, H is 2-degenerate, and the construction of H can be done in polynomial time. By Lemma 8, we have ${\tau }_C^{\left\{1\right\}}\left(G\right)\le k$ if and only if ${\tau }_C^{\left\{2\right\}}\left(H\right)\le k+6p.$ Note that ${\tau }_C^{\left\{1\right\}}\left(G\right)={\tau }_C\left(G\right)$, and the clique transversal problem is NP-complete when restricted to triangle-free, 2-degenerate, subcubic planar graphs by Theorem 4. Hence, the theorem holds.    □

Corollary 4.

Assume that d is a fixed integer larger than two. The $\left\{2\right\}$-clique transversal problem is NP-complete for d-degenerate graphs.

Proof. 

An ℓ-degenerate graph is d-degenerate if $\ell <d$. By Theorem 6, the corollary holds.    □

Theorem 7.

Let ℓ and b be fixed positive integers such that $\ell \ge 1$ and $d\ge 2$. The $\{2\ell \}$-clique transversal and the $\{2\ell +1\}$-clique transversal problems are NP-complete for $(d+1)$-degenerate graphs.

Proof. 

We can prove that the $\{2\ell \}$-clique transversal and the $\{2\ell +1\}$-clique transversal problems on $(d+1)$-degenerate graphs are in NP by an argument similar to that of the proof in Theorem 4.

Let G be a d-degenerate graph with $C\left(G\right)=\{C_i\mid 1\le i\le p\}$. Let $G_i$ be a $(5,1)$-tadpole graph with a $(5,1)$-tadpole ordering $(x_i^1,x_i^2,\dots ,x_i^6)$ for $1\le i\le p$. We construct H by (1) the union of G and each $G_i$ and (2) connecting $x_i^6$ to all vertices of $C_i$ for $1\le i\le p$. Clearly, H is $(d+1)$-degenerate, and the construction of H can be done in polynomial time. By Lemma 8, we have

(1)

${\tau }_C^{\{\ell \}}\left(G\right)\le k$ if and only if ${\tau }_C^{\{2\ell \}}\left(H\right)\le k+6p\ell$;

(2)

${\tau }_C^{\{\ell +1\}}\left(G\right)\le k$ if and only if ${\tau }_C^{\{2\ell +1\}}\left(H\right)\le k+(6\ell +3)·p.$

By Theorems 4 and 6, the $\left\{1\right\}$-clique transversal and the $\left\{2\right\}$-clique transversal problems are NP-complete for 2-degenerate graphs. Hence, the theorem holds.    □

6. Para-NP-Completeness Results

This section presents para-NP-completeness results for signed and minus clique transversal problem for d-degenerate graphs.

Definition 5.

A graph G is a $(2,\ell )$-split graph if $V\left(G\right)$ can be partitioned into a 2-clique Q and an independent set S of ℓ vertices such that every vertex in S is adjacent to all vertices in Q.

Figure 2 shows a $(2,3)$-split graph whose vertices have been partitioned into a 2-clique $Q=\{q_1,q_2\}$ and an independent set $S=\{s_1,s_2,s_3\}$.

Theorem 8

([16]).   ${\tau }_C^s\left(G\right)={\tau }_C^{-}\left(G\right)=2-\ell$ for any $(2,\ell )$-split graph.

Theorem 9.

  The signed clique transversal problem parameterized by the solution weight k is para-NP-complete for d-degenerate graphs with $d\ge 3$.

Proof. 

Corollary 2 reveals that the signed clique transversal problem is NP-complete for d-degenerate graphs with $d\ge 3$. Consequently, for any d-degenerate graph G and an arbitrary integer k (not necessarily fixed), the problem of finding a signed clique transversal function of G with weight of at most k is NP-complete. In the following, we prove the para-NP-completeness of the problem by showing that the problem parameterized by the solution weight k is NP-complete even when the parameter k is fixed.

Consider the zero-signed clique transversal problem for d-degenerate graphs with $d\ge 3$. The problem is a particular case of the signed clique transversal problem parameterized by the solution weight k when $k=0$. The zero-signed clique transversal problem on d-degenerate graphs with $d\ge 3$ is in NP by an argument similar to that of the proof in Theorem 4.

Given an arbitrary integer k and a fixed integer $d\ge 3$, we define G as a d-degenerate graph and $\widehat{G}$ as a $(2,k+2)$-split graph. Theorem 8 indicates that ${\tau }_C^s\left(\widehat{G}\right)=-k$. Let $H=G\cup \widehat{G}$. Clearly, ${\tau }_C^s\left(H\right)={\tau }_C^s\left(G\right)-k$ and H is d-degenerate. Hence, ${\tau }_C^s\left(G\right)\le k$ if only if ${\tau }_C^s\left(H\right)\le 0$. Therefore, the zero-signed clique transversal problem is NP-complete for d-degenerate graphs with $d\ge 3$. Recall that the problem is a particular case of the signed clique transversal problem parameterized by the solution weight k even when restricted to a fixed k. The theorem thus holds.    □

Theorem 10.

  The minus clique transversal problem parameterized by the solution weight k is para-NP-complete for d-degenerate graphs with $d\ge 2$.

Proof. 

Corollary 1 reveals that the minus clique transversal problem is NP-complete for d-degenerate graphs with $d\ge 2$. Consequently, for any d-degenerate graph G and an arbitrary integer k (not necessarily fixed), the problem of finding a minus clique transversal function of G with a weight of at most k is NP-complete. We can prove the para-NP-completeness of the problem by an argument similar to that of the proof in Theorem 9.    □

7. Fixed-Parameter Tractable Results

This section shows that the b-fold clique transversal and the $\left\{b\right\}$-clique transversal problems are fixed-parameter tractable for d-degenerate graphs.

Lemma 9.

Consider the b-fold clique transversal problem parameterized by the solution size k for a d-degenerate graph G. The following statements are true.

(1) 

If $b=1$, then every b-fold clique transversal set of G with a size of at most k contains all the vertices in the set $S=\{v\mid v\in V\left(G\right),de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)\}$, and $\left|S\right|\le k$.

(2) 

If $b\ge 2$ and G contains a vertex of a degree larger than $k·\left(\omega \right(G)-1)-1$, then there does not exist a b-fold clique transversal set with a size of at most k in G.

Proof. 

(1) Assume that $b=1$. Let v be a vertex in S. Then, $de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)$. There exist at least $\frac{de{g}_G\left(v\right)}{\omega \left(G\right)-1}$ maximal cliques in G such that all of them intersect only at v and have no other common edges. Note that $k+1=\frac{de{g}_G\left(v\right)}{\omega \left(G\right)-1}$. Let $C_1,\dots ,C_{k+1}$ be these maximal cliques. Assume that a b-fold clique transversal set D of G with $\left|D\right|\le k$ does not include the vertex v. Then, $|(C_i\backslash \left\{v\right\})\cap D|=|C_i\cap D|\ge b$ for $1\le i\le k+1$. The size of D is larger than or equal to $k+1$, which contradicts that $\left|D\right|\le k$. Therefore, D contains v. This implies that every b-fold clique transversal of G with a size of at most k contains every vertex in the set $S=\{v\mid v\in V\left(G\right),de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)\}$. Obviously, the set S is a subset of D. We have $\left|S\right|\le \left|D\right|\le k$. Statement (1) holds by the above discussion.

(2) Assume that $b\ge 2$. Let x be a vertex in G such that $de{g}_G\left(x\right)>k·(\omega \left(G\right)-1)-1$. Then, $de{g}_G\left(x\right)\ge k·(\omega \left(G\right)-1)$. Clearly, there are at least k maximal cliques $C_1,\dots ,C_k$ in G that intersect only at v and have no other common edges. Let D be a b-fold clique transversal set of G. We consider the following two cases.

Case 1: $x\notin D$. Then, $|(C_i\backslash \left\{x\right\})\cap D|=|C_i\cap D|\ge b\ge 2$ for $1\le i\le k$. The size of D is larger than or equal to $2k$. We have $\left|D\right|>k$. This leads to a contradiction.

Case 2: $x\in D$. Note that $b\ge 2$. Then, $\left|\right(C_i\backslash \left\{v\right\}\cap D|\ge (b-1)\ge 1$ for $1\le i\le k$. Consequently, $\left|D\right|\ge k+1$. This also leads to a contradiction. Statement (2) therefore holds by the above discussion.    □

Lemma 10.

Consider the b-fold clique transversal problem parameterized by the solution size k for a d-degenerate graph G. There is a problem kernel of size $O\left(k^2\right)$ for this parameterized problem.

Proof. 

We consider the following two cases according the value of b.

(1) Case 1: b=1. Let $S=\{v\in V\left(G\right)|de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)\}$, and $L=\{C\mid C\in C(G),C\cap S=\varnothing \}$. Let H be the clique union graph obtained from L. We have the following claim.

Claim 1.

G has a 1-fold clique transversal set of with a size of no more than k if and only if there exists a 1-fold clique transversal set $D^{\prime }$ of H such that $|D^{\prime }|\le k-|S|$.

Proof. 

Suppose that D is a 1-fold clique transversal set of G with $\left|D\right|\le k$. By Lemma 9, S is a subset set of D. The set $D^{\prime }=D\backslash S$ contains at least one vertex in C for each $C\in L$. Therefore, $D^{\prime }$ is a 1-fold clique transversal set of H with $|D^{\prime }|\le k-|S|$.

Conversely, we suppose that $D^{\prime }$ is a 1-fold clique transversal set of H with $|D^{\prime }|\le k-|S|$. Since S includes at least one vertex in C for each $C\in C\left(G\right)\backslash L$, the set $D=D^{\prime }\cup S$ is a 1-fold clique transversal set of G with $\left|D\right|\le k$. Hence, the claim holds from the above discussion.    □

Let $\kappa =k-\left|S\right|$. Assume that $D_H$ is a 1-fold clique transversal set of H with $|D_H|\le \kappa$. The set $D_H$ includes at least one vertex from each maximal clique in H. Therefore, the union of all cliques in $C\left(H\right)$ is equivalent to the union of the closed neighborhoods of all vertices in $D_H$. Since H does not contain any vertex in S, by Lemma 9, the degree of every vertex in H is no more than $(k+1)·\left(\omega \right(G)-1)-1$. We have

$$\begin{array}{cc}\left|V\right(H\left)\right|\hfill & =|\bigcup _{v\in D_H}{N}_H\left[v\right]|\le |D_H|+|\bigcup _{v\in D_H}{N}_H\left(v\right)|\hfill \\ & \le \kappa +\kappa ·\left((k+1)·\left(\omega \right(G)-1)-1\right)\hfill \\ & =\kappa (k+1)·\left(\omega \right(G)-1)\hfill \\ & \le k(k+1)·\left(\omega \right(G)-1).\hfill \end{array}$$

Following Lemma 1, we have $\omega \left(G\right)-1\le d$. Recall that d is fixed. Hence, $\left|V\left(H\right)\right|=O\left(k^2\right)$.

Conversely, we assume that $\left|V\right(H\left)\right|>k(k+1)·\left(\omega \right(G)-1)$. Let $D_H$ be a 1-fold clique transversal set of H with $|D_H|=\beta$. We have

$$\begin{array}{cc}\left|V\right(H\left)\right|\hfill & =|\bigcup _{v\in D_H}{N}_H\left[v\right]|\le |D_H|+|\bigcup _{v\in D_H}{N}_H\left(v\right)|\hfill \\ & \le \beta +\beta ·\left((k+1)·\left(\omega \right(G)-1)-1\right)\hfill \\ & =\beta (k+1)\left(\omega \right(G)-1).\hfill \end{array}$$

Consequently, $\beta >k\ge \kappa$. This implies that H does not have any 1-fold clique transversal set with a size of at most $\kappa$ if $\left|V\right(H\left)\right|>k(k+1)·\left(\omega \right(G)-1)$. Therefore, $\left|V\left(H\right)\right|=O\left(k^2\right)$. Note that the number of edges in a d-degenerate graph H is at most $d\left|V\left(H\right)\right|-(d^2+d)/2$ [2]. Then,

$$\begin{array}{cc}\left|E\right(H\left)\right|\hfill & \le d\left|V\left(H\right)\right|-(d^2+d)/2\hfill \\ & \le kd(k+1)·(\omega \left(G\right)-1)-(d^2+d)/2.\hfill \end{array}$$

Recall that d is fixed and $\omega \left(G\right)-1\le d$. Hence, $\left|E\left(H\right)\right|=O\left(k^2\right)$. Following Claim 1 and the above discussion, we know that there is a problem kernel of size $O\left(k^2\right)$ for this parameterized problem.

(2) Case 2: $b\ge 2$. By Lemma 9, G does not have a vertex with a degree larger than $k·\left(\omega \right(G)-1)-1$ if there exists a b-fold clique transversal set with a size of at most k in G. Let D be a b-fold clique transversal set of G with $\left|D\right|\le k$. We have

$$\begin{array}{cc}\left|V\right(G\left)\right|\hfill & =|\bigcup _{v\in D}{N}_G\left[v\right]|\le |D|+|\bigcup _{v\in D}{N}_G\left(v\right)|\hfill \\ & \le k+k·\left(k·\left(\omega \right(G)-1)-1\right)\hfill \\ & =k^2·(\omega \left(G\right)-1)\hfill \\ & \le k^{2}d.\hfill \end{array}$$

Therefore, $\left|V\left(G\right)\right|=O\left(k^2\right)$. Note that the number of edges in a d-degenerate graph G is at most $d\left|V\left(G\right)\right|-(d^2+d)/2$ [2]. Then,

$$\begin{array}{cc}\left|E\right(G\left)\right|\hfill & \le d\left|V\left(G\right)\right|-(d^2+d)/2\hfill \\ & \le d·k^2·(\omega \left(G\right)-1)-(d^2+d)/2.\hfill \\ & \le k^{2}d-(d^2+d)/2.\hfill \end{array}$$

Therefore, $\left|E\left(G\right)\right|=O\left(k^2\right)$. Following the above discussion, we know that there is a problem kernel of size $O\left(k^2\right)$ for this parameterized problem.   □

Lemma 11.

  It takes linear time to find a problem kernel of size $O\left(k^2\right)$ for the b-fold clique transversal problem parameterized by the solution size k for a d-degenerate graph G.

Proof. 

We give Algorithm 1 to find a problem kernel of size $O\left(k^2\right)$ for the parameterized problem. The algorithm initiates empty sets S and L, and it sets ℓ to zero.

Algorithm 1 RKS(G, b, k)

1: $S\leftarrow \varnothing$; 2: $L\leftarrow \varnothing$; 3: $\ell \leftarrow 0$; 4: if ($b==1$) then 5:        for each $v\in V\left(G\right)$ do 6:             if $de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)$ then 7:                  $S\leftarrow S\cup \left\{v\right\}$; 8:             end if 9:       end for 10:      if $\left|S\right|>k$ then 11:           stop and return the infeasibility of the problem; 12:      else 13:            for each $C\in C\left(G\right)$ do 14:               for each $v\in C$ do 15:                  if $de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)$ then 16:                        $\ell \leftarrow \ell +1$; 17:                  end if 18:            end for 19:            if ($\ell ==0$) then 20:                 $L\leftarrow L\cup \left\{C\right\}$; 21:            else 22:                  $\ell \leftarrow 0$; 23:            end if 24:         end for 25:     end if 26:     $\kappa \leftarrow k-\left|S\right|$; 27:     return S, L, and $\kappa$; 28: else                                                         ▹ the case for $b\ge 2$ 29:       for each $v\in V\left(G\right)$ do 30:            if $de{g}_G\left(v\right)\ge k·(\omega \left(G\right)-1)$ then 31:               stop and return the infeasibility of the problem; 32:            end if 33:      end for 34:      $L\leftarrow C\left(G\right)$; 35:      return S, L, and k; 36: end if

(1) Assume $b=1$. During the execution of lines 5 to 9, the algorithm evaluates the degree of each vertex in G to check if the degree surpasses a given threshold. Vertices with degrees exceeding this threshold are added to the set S. The process ensures that by the end of the evaluation, S consists of every vertex whose degree is greater than or equal to $(k+1)\left(\omega \right(G)-1)$. Therefore, $S=\{v\in V\left(G\right)\mid de{g}_G\left(v\right)\ge (k+1)(\omega \left(G\right)-1)\}$. Note that the number of edges in a d-degenerate graph H is at most $dn-(d^2+d)/2$ [2]. Consequently, the computational time during the evaluation phase (lines 5 to 9) is as follows:

$$\sum _{v\in V\left(G\right)}(deg\left(v\right)+1)=n+2m\le n+2(dn-(d^2+d)/2)=O\left(n\right).$$

Statement (1) of Lemma 9 confirms that every 1-fold clique transversal set of G with a size of at most k must include all vertices in S. If $\left|S\right|>k$, then it is impossible for G to have a 1-fold clique transversal set with a size of at most k. Between lines 10 and 11, the algorithm evaluates this condition. If the condition is true, the algorithm concludes that it is infeasible to find a 1-fold clique transversal set of G with a size of at most k. On the contrary, if the condition is not satisfied, then $\left|S\right|\le k$ and the algorithm proceeds to generate the set L.

From lines 13 to 24, the algorithm verifies each maximal clique in G for any vertex v with $de{g}_G\left(v\right)\ge (k+1)·(\omega \left(G\right)-1)$. Every maximal clique without such vertices is added to L if $\ell =0$.

Eppstein et al. [19] demonstrated that a d-degenerate graph with n vertices contains $O\left(n\right)$ maximal cliques, which can all be enumerated in $O\left(n\right)$ time. Furthermore, Lemma 1 indicates that the maximum clique size of a d-degenerate graph is $d+1$. The overall time required to construct L remains $O\left(n\right)$.

Note that $\left|S\right|\le k$ if the algorithm does not terminate at line 11. The algorithm sets $\kappa =k-\left|S\right|$ and outputs S, L, and $\kappa$.

(2) Assume that $b\ge 2$. Statement (2) of Lemma 9 states that there does not exist a b-fold clique transversal set with a size of at most k in G if G contains a vertex with a degree larger than $k·\left(\omega \right(G)-1)-1$. If this condition is met between lines 30 and 31, it is infeasible to find a b-fold clique transversal set of G with a size of at most k. Otherwise, if the algorithm does not stop at line 31, L is set as $C\left(G\right)$ and the algorithm returns S, L, and k. The time to form L remains $O\left(n\right)$.

In conclusion, the overall running time of the algorithm is $O\left(n\right)$. Clearly, the clique union graph obtained by L represents the problem kernel and can be constructed in $O\left(n\right)$. By Lemmas 9 and 10, the kernel is indeed of size $O\left(k^2\right)$. Hence, the lemma holds.    □

Theorem 11.

The 1-fold clique transversal problem parameterized by the solution size k can be solved in $O({(d+1)}^k{k}^2+n)$ time for a d-degenerate graph G.

Proof. 

We give Algorithm 2 to solve the parameterized problem for a d-degenerate graph G.

Algorithm 2 BFCS (S, L, k)

1: if $k=0$ and $L\ne \varnothing$ then 2:       return failure; 3: end if 4: if ($k\ge 0$ and $L=\varnothing$) then 5:       return S; 6: end if 7: if $k>0$ and $L\ne \varnothing$ then 8:       choose a maximal clique Q from L; 9:       $S^{\prime }\leftarrow \varnothing$; 10:      for each $x\in Q$ do 11:            $L^{\prime }\leftarrow \left\{C\right|C\in L,x\notin C\}$; 12:            $S^{\prime }\leftarrow BFCS(S\cup \left\{x\right\},L^{\prime },k-1)$; 13:            if $S^{\prime }\ne failure$ then 14:                  break; 15:            end if 16:     end for 17:     return $S^{\prime }$; 18: end if

The algorithm starts with the inputs S, L, and k derived from the output of Algorithm 1. The clique union graph obtained by L represents the problem kernel.

A d-degenerate graph G with n vertices contains $O\left(n\right)$ maximal cliques, which can all be enumerated in $O\left(n\right)$ time [19]. Furthermore, Lemma 1 indicates that the maximum clique size of G is $d+1$. By Lemma 10, there is a problem kernel of size $O\left(k^2\right)$ for the parameterization. Therefore, $\left|L\right|=O\left(k^2\right)$.

The algorithm attempts to find a subset S such that for each maximal clique in L, at least one element is chosen into S and the total number of selections does not exceed k. The algorithm uses a recursive, backtracking approach, effectively exploring different combinations of selections to find a valid solution. If it finds such a combination, it returns the subset; otherwise, it signifies failure.

We define $T\left(\right|L|,k)$ as the running time of Algorithm 2, which can be described by the following recurrence:

$$T\left(\right|L|,k)=\left\{\begin{array}{c}O\left(1\right)\mathrm{if} k=0 \mathrm{and} L\ne \varnothing ,\hfill \\ O\left(1\right)\mathrm{if} k\ge 0 \mathrm{and} L=\varnothing ,\hfill \\ (d+1)·\left(T\left(\right|L|-1,k-1)+O\left(k^2\right)\right)\mathrm{if} k>0 \mathrm{and} L\ne \varnothing .\hfill \end{array}\right.$$

Therefore, $T(L,k)=O\left({(d+1)}^k{k}^2\right)$. Following Lemma 11, the problem kernel can be found in $O\left(n\right)$ time.

Based on the discussion above, the total time needed to find a 1-fold clique transversal set with at most k vertices is $O({(d+1)}^k{k}^2+n)$. □

Theorem 12.

Assume that $b\ge 2$. The b-fold clique transversal problem parameterized by the solution size k can be solved in $O({(\omega \left(G\right)-b+0.2738)}^k{k}^{O\left(1\right)}+n)$ time for a d-degenerate graph G.

Proof. 

Given a universal set U and a family F of subsets of U, the HITTING SET problem is to find a minimum subset X of U such that $|X\cap S|\ge 1$ for every $S\in F$. If the size of every set in the family F is at most p, then the problem is known as the p-HITTING SET problem. Let $2\le \ell \le p$. The UNIFORM MULTI p-HITTING SET problem is to find a minimum subset X of U such that $|X\cap S|\ge \ell$ for every $S\in F$.

The b-fold clique transversal problem for a d-degenerate graph G is a particular case of the UNIFORM MULTI p-HITTING SET problem with $p=\omega \left(G\right)$, $U=V\left(G\right)$, $F=C\left(G\right)$, and $\ell =b$.

The UNIFORM MULTI p-HITTING SET problem parameterized by the solution size k can be solved in $O{(p-\ell +0.23738)}^k{n}^{O\left(1\right)})$ time, where n is the size of the given universal set [21]. By Lemmas 10 and 11, there is a problem kernel of size $O\left(k^2\right)$ for the b-fold clique transversal problem parameterized by the solution size k, and it can be found in linear time. Hence, the b-fold clique transversal problem parameterized by the solution size k can be solved in $O{(\omega \left(G\right)-b+0.2738)}^k{k}^{O\left(1\right)}+n)$ time for a d-degenerate graph G. □

Theorem 13.

The $\left\{b\right\}$-clique transversal problem parameterized by the solution weight k can be solved in $O{(b·\omega \left(G\right)-b+0.2738)}^k{k}^{O\left(1\right)}+n)$ time for a d-degenerate graph G.

Proof. 

Let G be a d-degenerate graph with $V\left(G\right)=\{v_i\mid 1\le i\le n\}$ and $C\left(G\right)=\{C_i\mid 1\le i\le \ell \}$. For each vertex $v_i\in V\left(G\right)$, let $V_i=\{v_i^1,v_i^2,\dots ,v_i^b\}$. For $1\le j\le \ell$, let $Q_j={\bigcup }_{v_i\in C_j}{V}_i$, and connect every two distinct vertices in $Q_j$. Let H be the clique union graph obtained from the set ${\bigcup }_{i=1}^{\ell }\left\{Q_i\right\}$. Clearly, $\omega \left(H\right)=b·\omega \left(G\right)$, and H can be constructed in linear time.

Consider an induced subgraph $H^{\prime }$ of H with at least two vertices. Let $V^{\prime }=\{v_j\mid V_j\cap V\left(H\right)\ne \varnothing ,1\le j\le n\}$. The graph $G\left[V^{\prime }\right]$ is an induced subgraph of G. Since G is a d-degenerate graph, there exists a vertex $v_r\in V^{\prime }$ such that the degree of $v_r$ in $G\left[V^{\prime }\right]$ is at most d. Therefore, there exists a vertex $v_r^q\in V\left(H\right)$ such that it is adjacent to at most $b·(d+1)$ vertices in $H^{\prime }$. Let $p=b·(d+1)$. Consequently, H is a p-degenerate graph.

Assume that f is a $\left\{b\right\}$-clique transversal function of G of minimum weight. Let $\widehat{V}=\{v\in V\left(G\right)\mid f\left(v\right)\ne 0\}$, and let $s_i=f\left(v_i\right)$ for each vertex $v_i\in \widehat{V}$. Let $S={\bigcup }_{v_i\in \widehat{V}}\{v_i^1,\dots v_i^{s_i}\}$. Let $C_i$ be a maximal clique in G. Then,

$$\sum _{v_j\in C_i}f\left(v_j\right)=\sum _{v_j\in C_i\cap \widehat{V}}f\left(v_j\right)=|\bigcup _{v_j\in C_i\cap \widehat{V}}\{v_j^1,\dots ,v_j^{s_j}\}|=|S\cap Q_i|\ge b.$$

Note that $C\left(H\right)=\{Q_1,\dots Q_{\ell }\}$. Therefore, S is a b-fold clique transversal set of H. We have ${\tau }_C^b\left(H\right)\le {\tau }_C^{\left\{b\right\}}\left(G\right)$.

Assume that D is a minimum b-fold clique transversal set of H. Let $\widehat{S}=\{v_i\mid D\cap V_i\ne \varnothing ,1\le i\le n\}$. Let $d_i=|D\cap V_i|$ for $1\le i\le n$. Let $g:V\left(G\right)\to \{0,1,\dots ,b\}$ be a function of G, and let $g\left(v_i\right)=d_i$ for $1\le i\le n$. Obviously, $g\left(v_j\right)=0$ if $v_j\notin \widehat{S}$. Let $Q_i$ be a maximal clique of $C\left(H\right)$. Then,

$$|D\cap Q_i|=|\bigcup _{j=1}^n(D\cap Q_i)\cap V_j|=|\bigcup _{v_j\in C_i\cap \widehat{S}}\{v_j^1,\dots ,v_j^{d_j}\}|=\sum _{v_j\in C_i\cap \widehat{S}}g\left(v_j\right)=\sum _{v_j\in C_i}g\left(v_j\right)\ge b.$$

Therefore, g is a $\left\{b\right\}$-clique transversal function of G. We have ${\tau }_C^{\left\{b\right\}}\left(G\right)\le {\tau }_C^b\left(H\right)$. Following the above discussion, we know that ${\tau }_C^{\left\{b\right\}}\left(G\right)={\tau }_C^b\left(H\right)$. Consequently, ${\tau }_C^{\left\{b\right\}}\le k$ if and only if ${\tau }_C^b\left(H\right)\le k$. By Theorem 12, the b-fold clique transversal problem parameterized by the solution size k can be solved in $O({(\omega \left(H\right)-b+0.2738)}^k{k}^{O\left(1\right)}+n)$ time for the p-degenerate graph H. Hence, the $\left\{b\right\}$-clique transversal problem parameterized by the solution size k can be solved in $O({(b·\omega \left(G\right)-b+0.2738)}^k{k}^{O\left(1\right)}+n)$ time for a d-degenerate graph G. □

8. Conclusions

This paper explored the computational aspects of various clique transversal problems on d-degenerate graphs, which are notably prevalent in the study of sparse graph classes. The theoretical investigations were centered on problems such as the b-fold and b-clique transversal problems and provided significant insights into their complexity and computational tractability. Our findings established a spectrum of computational complexities for these problems, ranging from polynomial-time solvability in specific graph classes to NP-completeness in more general settings. For instance, while certain cases of clique transversal problems are solvable in polynomial time when restricted to graphs like 1-degenerate graphs or graphs with bounded treewidth, they become NP-complete in broader classes such as d-degenerate graphs for $d\ge 2$. Particularly notable is the parameterized complexity analysis, which revealed that, while the problems are generally NP-complete, they become fixed-parameter tractable when parameterized by the solution size under certain conditions. This transition highlights the nuanced understanding required when dealing with different graph parameters and their impact on algorithmic design. The practical implications of this work are broad, especially in areas involving network analysis, database theory, and the design of algorithms for biological datasets where understanding the clique structure is crucial. Future work could extend these results to dynamic graph models or investigate approximation algorithms, providing a more comprehensive range of tools for tackling these complex problems in practical scenarios. This could open new pathways for employing clique transversal theory in real-world applications, enhancing our ability to manage and analyze complex networks effectively.

While our theoretical analysis provides significant insights, applying these results in real-world settings such as communication and large-scale social networks presents several challenges. Graphs in these settings are often inferred, incomplete, or very large, complicating direct application of our theoretical findings. Such complexities require adaptive methods and robust preprocessing techniques for missing or uncertain data. Moreover, the scalability of algorithms remains a critical concern; thus, future research should focus on developing scalable solutions and approximation methods that can efficiently handle large datasets. Addressing these practical constraints will be crucial for translating our theoretical advancements into valuable tools for network analysis and other applications.