Well-covered Graphs Research Articles

Operations on Well-Covered Graphs and the Roller-Coaster Conjecture

A graph $G$ is well-covered if every maximal independent set has the same cardinality. Let $s_k$ denote the number of independent sets of cardinality $k$, and define the independence polynomial of $G$ to be $S(G,z) = \sum s_kz^k$. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by $z^c$ where $c$ is a positive integer. We will apply power magnification to the recent Roller-Coaster Conjecture of Michael and Traves, proving in our main theorem that for sufficiently large independence number $\alpha$, it is possible to find well-covered graphs with the last $(.17)\alpha$ terms of the independence sequence in any given linear order. Also, we will give a simple proof of a result due to Alavi, Malde, Schwenk, and Erdős on possible linear orderings of the independence sequence of not-necessarily well-covered graphs, and we will prove the Roller-Coaster Conjecture in full for independence number $\alpha \leq 11$. Finally, we will develop two new graph operations that preserve well-coveredness and have interesting effects on the independence polynomial.

Weighted Well-Covered Graphs and Complexity Questions

A weighted graph G is called well-covered if all its maximal stable sets have the same weight. This common weight is the value of G. Let S be a stable set of G (possibly, $S = \emptyset$). The subgraph $G - N[S]$ is called a co-stable subgraph of G. We denote by Sub(G) the set of all co-stable subgraphs of G (considered up to isomorphism). A class of weighted graphs P is called co-hereditary if it is closed under taking co-stable subgraphs, i.e., $G \in \mathcal{P}$ implies ${\rm CSub}(G) \subseteq \mathcal{P}$. We note that the class $\mathcal{WW}$ of all weighted well-covered graphs is co-hereditary and characterize $\mathcal{WW}$ in terms of forbidden co-stable subgraphs. Then we use a reduction from Satisfiability to show that the following decision problems are NP-complete. Decision Problem 1 (Co-Stable Subgraph). Instance: A graph $G$ and a set $U \subseteq V(G)$ that induces a subgraph $H$. Question: Is $H$ a co-stable subgraph of $G$? Decision Problem 2 (Co-Stable Subgraph $H$). Instance: A graph $G$. Question: Is $H$ a co-stable subgraph of $G$? Let $\Delta(G)$ be the maximum vertex degree of a graph $G$. We show that recognizing weighted well-covered graphs with bounded $\Delta(G)$ can be done in polynomial time.

Well-located graphs: a collection of well-covered ones

A well-covered graph is one in which every maximal independent set is maximum. A dominating set, say D, is said to be locating if, for every pair of vertices not in the set D, their neighbours that are in D differ in at least one vertex. A graph is called well-located if it has the property that every independent dominating set is locating. In a previous paper (Congr. Numer. 65 (1988) 191–200) the authors showed that the well-located graphs are a subclass of the well-covered ones and, for girth 5, the two classes are identical. In this article, we begin the examination of the well-located graphs of girth 4 and, for those well-located graphs with no 4-cycles (but 3-cycles are allowed), obtain a characterization.

Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture

A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term s k of the independence sequence (s 0,s 1,?,s ?) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent unimodality conjecture of Brown, Dilcher, and Nowakowski. We formulate the Roller-Coaster Conjecture to describe the possible linear orderings of terms of the independence sequence.

Robust algorithms for restricted domains

We introduce a new definition of efficient algorithms for restricted domains. Under this definition, an algorithm is required to be “robust,” i.e., it must produce correct output regardless of whether the input actually belongs to the restricted domain or not. This is to be contrasted with the “promise” version of solving problems on restricted domains, in which there is a guarantee that the input is in the class, and an algorithm to “solve” the problem need not function correctly or even terminate if this guarantee is not met. There exist problems that have a polynomial time promise solution, while being NP-hard if required to be robust. We show perhaps the surprising result that robustly finding a maximum independent set in a well-covered graph (i.e., a graph in which every maximal independent set is of the same size) is NP-hard. An argument can be made that this hardness result is more meaningful than the trivial polynomial time promise algorithm. We give a polynomial time robust algorithm for the maximum clique problem in unit disk graphs, i.e., given an input graph G in general form, the output is either a maximum clique for G or a certificate that G is not a unit disk graph. The existence of this algorithm is to be reconciled with the apparent contradiction posed by the facts: (1) Recognizing whether an input graph given in general form is a unit disk graph is NP-hard; in fact, it is not even known to be in NP. (2) Finding a maximum clique in an input graph given in general form is NP-hard.

On well-covered graphs of odd girth 7 or greater

On well-covered graphs of odd girth 7 or greater

Characterization of [formula omitted]-well-covered graphs for some classes of graphs

A graph G is a Z m - well-covered graph if |I 1|≡|I 2|( mod m ) for all maximal independent sets I 1 and I 2 in V(G) [3]. The recognition problem of Z m -well-covered graphs is a Co-NP-Complete problem. We give a characterization of Z m -well-covered graphs for chordal, simplicial and circular arc graphs.

Well-located graphs: A collection of well-covered ones

A well-covered graph is one in which every maximal independent set is maximum. A dominating set, say D, is said to be locating if, for every pair of vertices not in the set D, their neighbours that are in D di/er in at least one vertex. A graph is called well-located if it has the property that every independent dominating set is locating. In a previous paper (Congr. Numer. 65 (1988) 191–200) the authors showed that the well-located graphs are a subclass of the well-covered ones and, for girth 5, the two classes are identical. In this article, we begin the examination of the well-located graphs of girth 4 and, for those well-located graphs with no 4-cycles (but 3-cycles are allowed), obtain a characterization. c 2003 Elsevier B.V. All rights reserved.

Solving Problems on Special Classes of Graphs

Abstract Most papers which solve a problem on a graph class assume there is a promise that the input is in the class. Thus, the algorithm could produce incorrect output, or fail to terminate in the time bound claimed, if the input is not in the class. We propose a stricter notion of solving problems on special classes of graphs. If a graph G is not in the class, the algorithm may either answer the problem correctly, or announce that G is not in the class. We call such an algorithm a robust algorithm. This talk describes robust algorithms for problems on classes where no robust algorithm was known previously, shows that some problems which can be solved on classes in polynomial time given a promise that the graph is in the class are intractable for robust algorithms, and poses open problems. We show that the clique problem on unit disk graphs can be solved robustly. Previous solutions needed a geometric model as input. We also show that no robust polynomial algorithm solves independent set on well-covered graphs unless P = NPM.

A characterization of Z m -well-covered graphs of girth 6 or more

The class of Zm-well-covered graphs, those in which the cardinality of every maximal independent subset of vertices is congruent to the same number modulo m, contains the well-covered graphs as well as parity graphs. Here we consider such graphs, where there is no small cycle present and obtain a characterization for those of girth 6 or more. Weighted well-covered graphs of girth 7 or more are also characterized, and this result is used, in turn, to characterize magic well-covered graphs of girth 7 or more. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 246–255, 2000

A characterization of well-covered graphs in terms of prohibited costable subgraphs

A graph is calledwell-covered if all maximal independent subsets in it have the same number of elements. LetI be an independent subset (possibly empty) in a graphG. The subgraph ofG obtained by erasing the setI together with its neighborhood is calledcostable. We present a characterization of the class of well-covered graphs in terms of the minimal set of prohibited costable subgraphs. This characterization implies characterizations of some well-known subclasses of the class of well-covered graphs as well as the existence of a polynomial algorithm for the recognition of well-covered graphs with bounded valences.

ModelingK-coteries by well-covered graphs

The concept of k-coterie is useful for achieving k-mutual exclusion in distributed systems. A graph is said to be well covered if any of its maximal independent sets is also maximum. We first show that a graph G is well covered with independence number k if and only if G represents the incidence relation among quorums forming a k-coterie. We then discuss the problem of constructing k-coteries having some desirable properties. We also characterize the well-covered graphs with independence number 2. © 1999 John Wiley & Sons, Inc. Networks 34: 221–228, 1999

Local Structure When All Maximal Independent Sets Have Equal Weight

In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal independent sets have equal weight. If a set is independent precisely when its elements are pairwise independent, a graph can be used as a model. The question then becomes whether a graph, with its vertices weighted by elements of an abelian group, is well-covered, i.e., has all maximal independent sets of vertices with equal weight. This problem is known to be co-NP-complete in general. We show that whether a graph is well-covered or not depends on its local structure. Based on this, we develop an algorithm to recognize well-covered graphs. For graphs with n vertices and maximum degree $\Delta$, it runs in linear time if $\Delta$ is bounded by a constant, and in polynomial time if $\Delta = O(\root 3 \of {\log n})$. We mention various applications to areas including hypergraph matchings and radius k independent sets. We extend our results to the problem of determining whether a graph has a weighting which makes it well-covered.

Subdivisions, parity and well-covered graphs

A graph is well-covered if every maximal independent set is maximum. This concept, introduced by Plummer in 1970 (J. Combin. Theory 8 (1970)), is the focal point of much interest and current research. We consider well-covered 2-degenerate graphs and supply a structural (and polynomial time algorithm) characterization of the class called 3-separable graphs. Also we consider parity graphs studied by Finbow and Hartnell and answer the question posed by them (Ars. Combin. 40 (1995)) by proving, among other results, that the decision problem: “given a graph G which is a parity graph, is G also well-covered graph?” is in the class CO-NPC. In addition we supply some complexity results that answer some problems due to Plummer (Quaestiones Math. 16 (1993)) and Finbow, Hartnell, and Whitehead (Discrete Math. 125 (1994)). © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 85–94, 1997

The Structure of Well-Covered Graphs and the Complexity of Their Recognition Problems

A graph is well-covered if all its maximal independent sets are of the same cardinality. Deciding whether a given graph is well-covered is known to be NP -hard in general, and solvable in polynomial time, if the input is restricted to certain families of graphs. We present here a simple structural characterization of well-covered graphs and then apply it to the recognition problem. Apparently, polynomial algorithms become easier to design. In particular we present a new polynomial time algorithm for the case where the input graph contains no induced subgraph isomorphic to K 1, 3 . Considering the line-graph of an input graph, this result provides a short and simple alternative to a proof by Lesk, Plummer and Pulleyblank, who showed that equimatchable graphs can be recognized in polynomial time.

Tracking the P‐NP boundary for well‐covered graphs

Tracking the P‐NP boundary for well‐covered graphs

Tracking the P-NP boundary for well-covered graphs

Certain fundamental graph problems like recognition, dominating set, Hamiltonian cycle and path, and clique partition, which are hard for well-covered graphs in general, can be solved efficiently for very well covered graphs. We address the question of how far the class of very well covered graphs can be generalized before such problems become hard. In this context, we examine the complexity of such problems for two subclasses of well-covered graphs, one properly containing the other, with the smaller one properly containing the family of very well covered graphs. We show that the problems studied algorithmically separate the subclasses from each other and from the families of well-covered and very well covered graphs.

Recursively decomposable well-covered graphs

We give an alternative characterization for well-covered graphs and restrict this to a characterization for very well-covered graphs. We state the conditions under which the intersection of a pair of maximal independent sets of a well-covered graph is maximal and use this result to define and characterize two recursively decomposable subclasses of well-covered graphs, one properly containing the other. We show that the smaller subclass, in turn, properly contains the family of very well covered graphs without isolated vertices.

On 4-connected claw-free well-covered graphs

Recognition of well-covered graphs is co-NP-complete, even when the graphs in question are k 1,4-free. On the other hand, very recently, an algorithm has been found which recognizes k 1,3-free well-covered graphs in polynomial time. Claw-free well-covered graphs having no 4-cycles have recently been characterized. In the present paper, we determine two different classes of claw-free well-covered graphs — those which are 4-connected and 4-regular as well as those which are 4-connected and planar.

Strongly well-covered graphs

A graph is well-covered if every maximal independent set is a maximum independent set. A strongly well-covered graph G has the additional property that G- e is also well-covered for every line e in G. Hence, the strongly well-covered graphs are a subclass of the well-covered graphs. We characterize strongly well-covered graphs with independence number two and determine a parity condition for strongly well-covered graphs with independence number three. More generally, we show that a strongly well-covered graph (with more than four points) is 3-connected and has minimum degree at least four.