Well-covered Graphs Research Articles

Very Well-Covered Graphs via the Rees Algebra

Very Well-Covered Graphs via the Rees Algebra

Characterizations of Minimal Dominating Sets in γ-Endowed and Symmetric γ-Endowed Graphs with Applications to Structure-Property Modeling

Claude Berge (1987) introduced the concept of k-extendable graphs, wherein any independent set of size k is inherently a constituent of a maximum independent set within a graph H=(V,E). Graphs possessing the property of being 1-extendable are termedas Berge graphs. This introduction gave rise to the notion of well-covered graphs and well-dominated graphs. A graph is categorized as well-covered if each of its maximal independent sets is, in fact, a maximum independent set. Similarly, a graph attains the classification of well-dominated if every minimal dominating set (DS) within it is a minimum dominating set. In alignment with the concept of k-extendable graphs, the framework of (k,γ)-endowed graphs and symmetric (k,γ)-endowed graphs are established. In these graphs, each DS of size k encompasses a minimum DS of the graph. In this article, a study of γ-endowed dominating sets is initiated. Various results providing a deep insight into γ-endowed dominating sets in graphs such as those characterizing the ones possessing a unique minimum DS are proven. We also introduce and study the symmetric γ-endowed graphs and minimality of dominating sets in them. In addition, we give a solution to an open problem in the literature. which seeks to find a domination-based parameter that has a correlation coefficient of ρ>0.9967 with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the upper dominating number Γ(H) studied in this paper delivers a strong prediction potential.

Recognizing W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} Graphs

Let G be a graph. A set S⊆V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\subseteq V(G)$$\\end{document} is independent if its elements are pairwise nonadjacent. A vertex v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v \\in V(G)$$\\end{document} is shedding if for every independent set S⊆V(G)\\N[v]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\subseteq V(G) {\\setminus } N[v]$$\\end{document} there exists u∈N(v)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u \\in N(v)$$\\end{document} such that S∪{u}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\cup \\{u\\}$$\\end{document} is independent. An independent set S is maximal if it is not contained in another independent set. An independent set S is maximum if the size of every independent set of G is not bigger than |S|. The size of a maximum independent set of G is denoted α(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha (G)$$\\end{document}. A graph G is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is α(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha (G)$$\\end{document}. The graph G belongs to class W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} if every two disjoint independent sets in G are included in two disjoint maximum independent sets. If a graph belongs to the class W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document}, then it is well-covered. Finding a maximum independent set in an input graph is a well-known NP-hard problem. Recognizing well-covered graphs is co-NP-complete. Recently, it was proved that deciding whether an input graph belongs to the class W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} is co-NP-complete. However, when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} is still not known. In this article, we investigate the connection between shedding vertices and W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} graphs. On the one hand, we prove that recognizing shedding vertices is co-NP-complete. On the other hand, we find polynomial solutions for restricted cases of the problem. We also supply polynomial characterizations of several families of W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} graphs.

Computing well-covered vector spaces of graphs using modular decomposition

A graph is well-covered if all its maximal independent sets have the same cardinality. This concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph G, a real-valued vertex weight function w is said to be a well-covered weighting of G if all its maximal independent sets are of the same weight with respect to w. The set of all well-covered weightings of a graph G forms a vector space over the field of real numbers, called the well-covered vector space of G. Since the problem of recognizing well-covered graphs is co\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{co}$$\\end{document}-NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-complete, the problem of computing the well-covered vector space of a given graph is co\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{co}$$\\end{document}-NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graphs. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies a polynomial-time recognition algorithm for the class of well-covered fork-free graphs and also generalizes some known results on cographs.

Lower bounds for the depth of symbolic powers of edge ideals of graphs

Assume that G is a graph with edge ideal I(G). For every integer s≥1, we denote the s-th symbolic power of I(G) by I(G)(s). It is shown that for every very well-covered graph G (without isolated vertices) and for each integer s≥1, we have depthS/I(G)(s)≥2c(G)−p(G), where c(G) and p(G) denote the number of connected components and the number of bipartite connected components of G, respectively. Moreover, for any graph G, we determine a sharp lower bound for the depth of I(G)(3) in terms of the star packing number of G.

On the Castelnuovo–Mumford regularity of squarefree powers of edge ideals

Assume that G is a graph with edge ideal I(G) and matching number match(G). For every integer s≥1, we denote the s-th squarefree power of I(G) by I(G)[s]. It is shown that for every positive integer s≤match(G), the inequality reg(I(G)[s])≤match(G)+s holds provided that G belongs to either of the following classes: (i) very well-covered graphs, (ii) semi-Hamiltonian graphs, or (iii) sequentially Cohen-Macaulay graphs. Moreover, we prove that for every Cameron-Walker graph G and for every positive integer s≤match(G), we have reg(I(G)[s])=match(G)+s.

Recognizing well-dominated graphs is coNP-complete

A graph G is well-covered if every minimal vertex cover of G is minimum, and it is well-dominated if every minimal dominating set of G is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were first introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs has been proved to be coNP-complete by [Chvátal and Slater, AODM 1993] and [Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing well-dominated graphs has been left open since their introduction. We close this complexity gap by proving that recognizing well-dominated graphs is coNP-complete. This solves a well-known open question (cf. [Levit and Tankus, DM 2017] and [Gözüpek, Hujdurovic and Milanič, DMTCS 2017]), which was first asked in [Caro, Sebő and Tarsi, JAlg 1996]. Although the problem has been open for a long time, our proof is surprisingly simple. Finally, we show that recognizing well-totally-dominated graphs is coNP-complete, answering a question of [Bahadır, Ekim, and Gözüpek, AMC 2021].

Very well-covered graphs with the Erdős–Ko–Rado property

A family of independent $r$-sets of a graph $G$ is an $r$-star if every set in the family contains some fixed vertex $v$. A graph is $r$-EKR if the maximum size of an intersecting family of independent $r$-sets is the size of an $r$-star. Holroyd and Talbot conjecture that a graph is $r$-EKR as long as $1\leq r\leq\frac{\mu(G)}{2}$, where $\mu(G)$ is the minimum size of a maximal independent set. It is suspected that the smallest counterexample to this conjecture is a well-covered graph. Here we consider the class of very well-covered graphs $G^*$ obtained by appending a single pendant edge to each vertex of $G$. We prove that the pendant complete graph $K_n^*$ is $r$-EKR when $n \geq 2r$ and strictly so when $n>2r$. Pendant path graphs $P_n^*$ are also explored and the vertex whose $r$-star is of maximum size is determined.

Partitions and well-coveredness: The graph sandwich problem

A graph G is well-covered if every maximal independent set of G is maximum. A (k,ℓ)-partition of a graph G is a partition of its vertex set into k independent sets and ℓ cliques. A graph is (k,ℓ)-well-covered if it is well-covered and admits a (k,ℓ)-partition. The recognition of (k,ℓ)-well-covered graphs is polynomial-time solvable for the cases (0,1), (0,2), (1,0), (1,1), (1,2), and (2,0), and hard, otherwise. In the graph sandwich problem for a property Π, we are given a pair of graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2, and asked whether there is a graph G=(V,E) with E1⊆E⊆E2, such that G satisfies the property Π. The problem of recognizing whether a graph G satisfies a property Π is equivalent to the particular graph sandwich problem where E1=E2. In this paper, we study the graph sandwich problem for the property of being (k,ℓ)-well-covered. We present some structural characterizations and extending previous studies on the recognition of (k,ℓ)-well-covered graphs, we prove that Graph Sandwich for(k,ℓ)-well-coveredness is polynomial-time solvable when (k,ℓ)∈{(0,1),(1,0),(1,1),(0,2)}. Besides, we show that it is NP-complete for the property of being (1,2)-well-covered.

Well-hued graphs

We define a graph G as well-hued if for each positive integer k there is an integer a k such that every maximal k -colorable subgraph of G has order a k . This strengthens the notion of well-covered graphs. In this paper we investigate the connection between this property and related properties, characterize the sequences of the a k that can be achieved, and determine conditions for the property in some graph classes and under some graph operations such as products.

On the probe problem for (r,ℓ)-well-coveredness: Algorithms and complexity

Let C be a class of graphs. A graph G=(V,E) is C-probe if V(G) can be partitioned into two sets: non-probesN and probesP, where N is an independent set and new edges may be added between some non-probe vertices such that the resulting graph is in the class C. In this case, we say that (N,P) is a C-probe partition of G. In the Unpartitioned Probe problem for a graph class C we are given a graph G and asked whether G has a C-probe partition, i.e., such a problem consist of recognizing the class of C-probe graphs. A graph G=(V,E) is an (r,ℓ)-graph when V can be partitioned into (S1,S2,…,Sr,K1,K2,…,Kℓ) such that S1,S2,…,Sr are independent sets, and K1,K2,…,Kℓ are cliques. A graph G is well-covered if every maximal independent set is also maximum, and it is (r,ℓ)-well-covered if it is well-covered as well as an (r,ℓ)-graph. In this paper, we study the complexity of the Unpartitioned Probe problem for the class of (r,ℓ)-well-covered graphs. We classify all but the (2,0) and (1,2) cases.

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra

Very well-covered graphs and local cohomology of their residue rings by the edge ideals

In this paper, we deal with very well-covered graphs. We first describe the structure of these kinds of graphs based on the structure of Cohen–Macaulay very well-covered graphs. As an application, we analyze the structure of local cohomology of the residue rings by the edge ideals of very well-covered graphs. Also, we give different formulas of regularity and depth of these rings from known ones and we finally treat the CMt property.

A Survey of Unmixed Bipartite Graphs and Very Well-Covered Graphs

Many authors investigated unmixed and bipartite graphs. There are many results obtained on these two types of graphs and show that a bipartite graph G satisfies Serre’s condition S2, then G is Cohen-Macaulay. This paper is a survey with some new results about very well-covered graphs, an unmixed bipartite graph.

Weighted well-covered graphs without cycles of lengths 5, 6 and 7

A graph G is well-covered if all maximal independent sets are of the same cardinality. Assume that a weight function w:V(G)⟶R is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space[2] denoted WCW(G)[1]. Deciding whether an input graph G is well-covered is co-NP-complete [4,17]. Therefore, finding WCW(G) is co-NP-hard. This paper presents an algorithm whose input is a graph G without cycles of lengths 5, 6, and 7. The algorithm finds polynomially WCW(G).

Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs that have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes, and Toledo.

Linear polynomials for the regularity of powers of edge ideals of very well-covered graphs

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. We prove that if G is a very well-covered graph then for all s≥1 the regularity of I(G)s is exactly 2s+ν(G)−1, where ν(G) denotes the induced matching number of G.

Well-covered token graphs

The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs $G$, we classify when $T_k(G)$ is well-covered. For an arbitrary graph $G$, we show that if $T_2(G)$ is well-covered, then the girth of $G$ is at most four. We include upper and lower bounds on the independence number of $T_k(G)$, and provide some families of well-covered token graphs.

Mind the independence gap

The independence gap of a graph was introduced by Ekim et al. in 2018 as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set.We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomial-time algorithm for recognizing graphs of constant independence gap in any class of perfect graphs of bounded clique number. Next, we introduce a hereditary variant of the parameter, which we call hereditary independence gap and which measures the maximum independence gap over all induced subgraphs of the graph. We show that determining whether a given graph has hereditary independence gap at most k is polynomial-time solvable if k is fixed and co-NP-complete if k is part of input. We also investigate the complexity of the independent set and independent domination problems in graph classes related to independence gap. In particular, we show that the two problems are NP-complete in the class of graphs of independence gap at most one and that the weighted independent set problem is polynomial-time solvable in any class of graphs with bounded hereditary independence gap. Combined with some known results on claw-free graphs, our results imply that the independent domination problem is solvable in polynomial time in the class of {claw, 2P3}-free graphs.

On the structure of 4-regular planar well-covered graphs

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. It is unknown whether membership in this class of graphs is polynomially decidable. In this paper we focus on the study of well-covered graphs which are 4-regular and planar.