Well-covered Graphs Research Articles

Graphs in which all maximal bipartite subgraphs have the same order

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered.

On well-covered direct products

A graph $G$ is well-covered if all maximal independent sets of $G$ have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have nontrivial factorizations with respect to some of the standard graph products. In particular, they showed that both factors of a well-covered direct product are also well-covered and proved that the direct product of two complete graphs (respectively, two cycles) is well-covered precisely when they have the same order (respectively, both have order 3 or 4). Furthermore, they proved that the direct product of two well-covered graphs with independence number one-half their order is well-covered. We initiate a characterization of nontrivial, connected well-covered graphs $G$ and $H$, whose independence numbers are strictly less than one-half their orders, such that their direct product $G \times H$ is well-covered. In particular, we show that in this case both $G$ and $H$ have girth 3 and we present several infinite families of such well-covered direct products. Moreover, we show that if $G$ is a factor of any well-covered direct product, then $G$ is a complete graph unless it is possible to create an isolated vertex by removing the closed neighborhood of some independent set of vertices in $G$.

Detecting strong cliques

A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their complexity in the classes of chordal graphs, weakly chordal graphs, line graphs and their complements, and graphs of maximum degree at most three. Our results rely on connections with matchings and relate to several graph properties studied in the literature, including well-covered graphs, localizable graphs, and general partition graphs.

Shedding vertices of vertex decomposable well-covered graphs

We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen–Macaulay graphs.

On Well-Covered Cartesian Products

In 1970, Plummer defined a well-covered graph to be a graph G in which all maximal independent sets are in fact maximum. Later Hartnell and Rall showed that if the Cartesian product $$G \Box H$$ is well-covered, then at least one of G or H is well-covered. In this paper, we consider the problem of classifying all well-covered Cartesian products. In particular, we show that if the Cartesian product of two nontrivial, connected graphs of girth at least 4 is well-covered, then at least one of the graphs is $$K_2$$ . Moreover, we show that $$K_2 \Box K_2$$ and $$C_5 \Box K_2$$ are the only well-covered Cartesian products of nontrivial, connected graphs of girth at least 5.

Graph Operations Preserving W2-Property

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. Staples showed a number of ways to build graphs in Wn, using graphs from Wn or Wn+1. In this paper, we construct some more infinite subfamilies of the class W2 by means of corona, join, and rooted product of graphs.

On the (parameterized) complexity of recognizing well-covered (r,ℓ)-graph

An (r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓ cliques. A graph is (r,ℓ) if it admits an (r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)-well-covered if it is both (r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)-Well-Covered Graph problem ((r,ℓ)wc-g for short), we are given a graph G, and the question is whether G is an (r,ℓ)-well-covered graph. In the Well-Covered(r,ℓ)-Graph problem (wc-(r,ℓ)g for short), we are given an (r,ℓ)-graph G together with an (r,ℓ)-partition, and the question is whether G is well-covered. This generates two infinite families of problems, for any fixed non-negative integers r and ℓ, which we classify as being P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc-(r,0)g for r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size α of a maximum independent set of the input graph, its neighborhood diversity, its clique-width, or the number ℓ of cliques in an (r,ℓ)-partition. In particular, we show that the parameterized problem of determining whether every maximal independent set of an input graph G has cardinality equal to k can be reduced to the wc-(0,ℓ)g problem parameterized by ℓ. In addition, we prove that both problems are coW[2]-hard but can be solved in XP-time.

Graphs vertex-partitionable into strong cliques

A clique in a graph is strong if it intersects all maximal independent sets. A graph is localizable if it has a partition of the vertex set into strong cliques. Localizable graphs were introduced by Yamashita and Kameda in 1999 and form a rich class of well-covered graphs that coincides with the class of well-covered graphs within the class of perfect graphs. In this paper, we give several equivalent formulations of the property of localizability and develop polynomially testable characterizations of localizable graphs within three non-perfect graph classes: triangle-free graphs, C4-free graphs, and line graphs. Furthermore, we use localizable graphs to construct an infinite family of counterexamples to a conjecture due to Zaare-Nahandi about k-partite well-covered graphs having all maximal cliques of size k.

1-well-covered graphs revisited

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of any one vertex leaves a graph, which is well-covered as well (Staples, 1975).A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G∈W2 if and only if it is a 1-well-covered graph without isolated vertices.We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given independent set. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs.

On the unimodality of independence polynomials of very well-covered graphs

The independence polynomial i(G,x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n∕2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of such polynomials.

The Roller-Coaster Conjecture Revisited

A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer in J Comb Theory 8:91–98, 1970). If G is a well-covered graph with at least two vertices, and $$G{-}v$$ is well-covered for every vertex v, then G is a 1-well-covered graph (Staples in On some subclasses of well-covered graphs. Ph.D. Thesis, Vanderbilt University, 1975). The graph G is $$\lambda $$ -quasi-regularizable if $$\lambda >0$$ and $$\lambda \cdot \vert S\vert \le \vert N( S) \vert $$ for every independent set S of G. It is known that every well-covered graph without isolated vertices is 1-quasi-regularizable (Berge in Ann Discret Math 12:31–44, 1982). The independence polynomial $$I(G;x)= {\sum _{k=0}^{\alpha }} s_{k}x^{k}$$ is the generating function of independent sets in a graph G (Gutman and Harary in Util Math 24:97–106, 1983), where $$\alpha $$ is the independence number of G. The Roller-Coaster Conjecture (Michael and Traves in Graphs Comb 19:403–411, 2003), saying that for every permutation $$\sigma $$ of the set $$\{\lceil \frac{\alpha }{2}\rceil ,\ldots ,\alpha \}$$ there exists a well-covered graph G with the independence number $$\alpha $$ such that the coefficients $$( s_{k}) $$ of I(G; x) satisfy $$\begin{aligned} s_{\sigma \left( \left\lceil \frac{\alpha }{2}\right\rceil \right) }<s_{\sigma \left( \left\lceil \frac{\alpha }{2}\right\rceil +1\right) }<\cdots <s_{\sigma (\alpha )}, \end{aligned}$$ has been validated in Cutler and Pebody (J Comb Theory A 145:25–35, 2017). In this paper we show that independence polynomials of $$\lambda $$ -quasi-regularizable graphs are partially unimodal. More precisely, the coefficients of an upper part of I(G; x) are in non-increasing order. Based on this finding, we prove that the unconstrained part of the independence sequence is: $$\begin{aligned} \left( s_{\left\lceil \frac{\alpha }{2}\right\rceil },s_{\left\lceil \frac{\alpha }{2}\right\rceil +1},\ldots ,s_{\min \left\{ \alpha ,\left\lceil \frac{n-1}{3}\right\rceil \right\} }\right) \end{aligned}$$ for well-covered graphs, and $$\begin{aligned} \left( s_{\left\lceil \frac{2\alpha }{3}\right\rceil },s_{\left\lceil \frac{2\alpha }{3}\right\rceil +1},\ldots ,s_{\min \left\{ \alpha ,\left\lceil \frac{n-1}{3}\right\rceil \right\} }\right) \end{aligned}$$ for 1-well-covered graphs, where $$\alpha $$ stands for the independence number, and n is the cardinality of the vertex set.

W2-graphs and shedding vertices

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (Staples, 1975).A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G∈W2 if and only if it is a 1-well-covered graph without isolated vertices.We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given A∈Ind(G) in a graph G, where Ind(G) is the family of all the independent sets. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered.

Reconfiguring dominating sets in some well-covered and other classes of graphs

Let G be a graph and let k be a positive integer. The k-dominating graph of G, denoted by Dk(G), has vertices corresponding to the dominating sets of G having cardinality at most k; two vertices of Dk(G) are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote by d0(G) the least value of k for which Dk(G) is connected for all k≥d0(G). It is known that d0(G)≥Γ(G)+1, where Γ(G) is the upper domination number of G. In this paper we prove that if G is a graph that is both perfect and irredundant perfect, or if G belongs to certain classes of well-covered graphs, then d0(G)=Γ(G)+1. In order to prove these results, we show that all independent dominating sets of a graph G are in the same component of DΓ(G)+1(G).

THE PROJECTIVE DIMENSION OF THE EDGE IDEAL OF A VERY WELL-COVERED GRAPH

A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen–Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise$3$-disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

Crowns in bipartite graphs

A set S⊆V(G) is stable (or independent) if no two vertices from S are adjacent. Let Ψ(G) be the family of all local maximum stable sets [V. E. Levit, E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discr. Appl. Math. 124 (2002) 91–101] of graph G, i.e., S∈Ψ(G) if S is a maximum stable set of the subgraph induced by S∪N(S), where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order |I|+|N(I)|, and we write I∈Crown(G) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430].In this paper we show that Crown(G)⊆Ψ(G) holds for every graph, while Crown(G)=Ψ(G) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.

Maximal-clique partitions and the Roller Coaster Conjecture

A graph G is well-covered if every maximal independent set has the same cardinality q. Let ik(G) denote the number of independent sets of cardinality k in G. Brown, Dilcher, and Nowakowski conjectured that the independence sequence (i0(G),i1(G),…,iq(G)) was unimodal for any well-covered graph G with independence number q. Michael and Traves disproved this conjecture. Instead they posited the so-called “Roller Coaster” Conjecture: that the termsi⌈q2⌉(G),i⌈q2⌉+1(G),…,iq(G) could be in any specified order for some well-covered graph G with independence number q. Michael and Traves proved the conjecture for q<8 and Matchett extended this to q<12.In this paper, we prove the Roller Coaster Conjecture using a construction of graphs with a property related to that of having a maximal-clique partition. In particular, we show, for all pairs of integers 0≤k<q and positive integers m, that there is a well-covered graph G with independence number q for which every independent set of size k+1 is contained in a unique maximal independent set, but each independent set of size k is contained in at least m distinct maximal independent sets.

Regularity and projective dimension of some class of well-covered graphs

In this paper we study the Castelnuovo--Mumford regularity of an edge ideal associated with a graph in a special class of well-covered graphs. We show that if $G$ belongs to the class $\mathcal {SQ}$, then the Castelnuovo-Mumford regularity of $R/I(G)$ will be equal to induced matching number of $G$. For this class of graphs we also compute the projective dimension of the ring $R/I(G)$. As a corollary we describe these invariants in well-covered forests, well-covered chordal graphs, Cohen-Macaulay Cameron-Walker graphs, and simplicial graphs.

Pure simplicial complexes and well-covered graphs

A graph G is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complexis called pure if all of its facets have the same cardinality. Let G be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, we prove that for any simplicial complex or any graph, there is a corresponding graph in class G with the same well-coveredness property. Then some necessary and sufficient conditions are presented t o recognize which graphs in the class G are well-covered. This characterization has a nice algebraic interpretation according to zero-divisor elements of edge ring of graphs which is shown in this paper.

Well-covered graphs without cycles of lengths 4, 5 and 6

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight, where a weight of a set is the sum of weights of its elements. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of lengths 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered.Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY. Assume that there exists an independent set S such that each of S∪BX and S∪BY is a maximal independent set of G. Then B is a generating subgraph of G, and it produces the restriction w(BX)=w(BY). It is easy to see that for every weight function w, if G is w-well-covered, then the above restriction is satisfied.In the special case, where BX={x} and BY={y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of lengths 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of lengths 5 and 6.

Very well-covered graphs and their h-vectors

Let G be a Cohen–Macaulay very well-covered graph. We prove that the h-vector of the independence complex of G is precisely the f-vector of a flag complex. Moreover, we show that the h-vector of clique-whiskered graphs is exactly the h-vector of Cohen–Macaulay very well-covered graphs. In particular, we show that Kalai’s conjecture holds if G is a very well covered Cohen–Macaulay graph.