No abstract

No abstract

No abstract

No 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