Well-Covered Graphs: A Survey

Abstract

D)A graph G is well-covered (or w-c) if every maximal independent set of points in G is also maximum. Clearly, this is equivalent to the property that the greedy algorithm for constructing a maximal independent set always results in a maximum independent set. Although the problem of independence number is well-known to be NP-complete, it is trivially polynomial for well-covered graphs.The concept of well-coveredness was introduced by the author in [P1] and was first discussed therein with respect to its relationship to a number of other properties involving the independence number.Since then, a number of results about well-covered graphs have been obtained. It is our purpose in this paper to survey these results for the first time. As the reader will see, many of the results we will discuss are quite recent and have not as yet appeared in print. Background, Terminology and an Updated Lattice of ImplicationsIn this paper, all graphs will be assumed to be connected. A set of points is independent if no two of its members are joined by a line. The size of a largest (i.e., maximum) independent set will be denoted by a(G) and this number will be called the independence number of G. A point cover for a graph G is a set S C V(G) such that every line of G has at least one endpoint in S. Let us denote the size of a smallest point cover in a graph G by r(G). It is clear that the complement of a point cover is an independent set (and vice-versa) and that in fact, the complement of a minimum point cover is a maximum .a independent set (and vice-versa). Thus we always have that a(G) + r(G) = IV(G)i. We a.-ithus have the option of adopting the point of view of independent sets or of point covers. oId Historically, well-covered graphs (the subject of this paper) were defined in terms of point covers (hence the name), but more recently, most researchers in the area have converted to the independent set point of view. We too will follow this more recent trend. In 1972, Karp [K] showed that the independence number problem for graphs in general is NP-complete. Thus it is considered most unlikely that a polynomial algorithm will ever 5.u.2 be found to compute a(G). 92-051189 2 2 27 oll!,l~llllllll

A characterization of triangle-free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals

“…In general, to compute α(G) of a graph G is an NPcomplete problem (see [10]), but it is polynomial for well-covered graphs. Characterize well-covered graphs is a difficult problem and the work on well-covered graphs that has appeared in the literature has focused on certain subclasses of well-covered graphs (see [15] for the survey).…”

“…Such graphs form the so-called class W 2 (see [20]). Thus in order to characterize Gorenstein graphs we should classify the class W 2 , but this problem is also difficult (see [15]). Note that the Gorenstein property of graphs is also dependent on the characteristic of the base field k (see Proposition 2.1), so we cannot graph-theoretically characterize all Gorenstein graphs.…”

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

“…This notion was introduced by Plummer in [6]. Several results on the subject have been published since then, and a thorough review can be found in [5].…”

“…Such results are proved, either by means of a complete, easy (i.e., polynomial) to detect, structural characterization of well-covered graphs in the family (such characterizations are presented and reviewed in [5]), or by an explicit description of a polynomial algorithm. Lesk et al [3] developed a polynomial time algorithm to recognize graphs in which all maximal matchings are maximum.…”

Buchsbaumness of the second powers of edge ideals

“…A well-covered graph G is a member of the class W 2 if the remove any vertex of G leaves a well-covered graph with the same independence number as G (see e.g. [14]). …”