A characterization of well covered block-cactus graphs

A graph G is well covered, if any two maximal independent sets of G have the same number of vertices. A graph is called a block-cactus graph if each block is complete or a cycle. In this paper we characterize the well covered block-cactus graphs. 1. TERMINOLOGY AND INTRODUCTION In this paper we consider finite, undirected, and simple graphs G with the vertex set V(G). The degree d(x, G) of a vertex x of G is the number of edges incident \vith x. We denote by ]{n the complete graph of order n. For A ~ V(G) let G[A] be the subgraph induced by A. Moreover, N(x, G) denotes the set of vertices adjacent to the vertex x and, more generally, N(X,G) = UXExN(x, G) for a subs'et X of V(G). We write N[x,G] and N[X,G] instead of N(x,G) U x and N(X,G) U X. A cycle of length n is denoted by en = XIX2"'XnXl' A vertex c of a graph G is called a cut vertex of G if G c has more components than: G. A connected graph with no cut vertex is called a block. A block of a graph G is a subgraph of G which is itself a block and which is maximal with respect to that property. A graph G is a block graph if every block of G is a complete graph. A graph G is called a block-cactus graph if every block is complete or a cycle. A set I ~ V (G) is an independent set of G, if N(x, G) n I = 0 for every x E I. Let i(G) and a(G) denote the minimum and maximum cardinality of a maximal independent set in G. A graph G is said to be well covered if every maximal independent set in G is a maximum independent set in G. Equivalently, G is well covered if i( G) = a( G). The concept of well covered graphs was introduced by Plummer [6] and studied in a few papers. In particular, the well covered bipartite graphs were characterized by Favaron Ravindra [9], and Staples [10]. The cubic, planar, and 3-connected well covered graphs have been characterized in [1] by Campbell and Plummer. Recently, Finbow, Hartnell, and Nowakowski [3] and Prisner, Topp, and Vestergaard [8] have described the well covered graphs of girth at least five, and the well covered simplicial and chordal graphs, respectively. Additional exarnples and properties of well covered g~aphs may be found in the survey paper of Plummer [7]. In this paper we characterize the well covered block-cactus graphs. 2. PRELIMINARY RESULTS The following simple property of well covered graphs was first observed by Campbell and Plummer [1]. Proposition 2.1 ([1]). If G is a well covered graph, then for each vertex v E V(G), the graph 0 G] is well covered. Proposition 2.2. If X and Yare two independent sets of a graph 0 with IXI > IYI and N[X, OJ ~ N[Y, GJ, then 0 is not well covered. Proof. Assume to the contrary that G is well covered. Then, for every maximal independent set I with Y ~ I, we have III = 0:( G). Furthermore, it follows from N[X, 0] ~ N[Y, 0] that J (1 Y)UX is an independent set such that a(O) 2: IJI = 1(1 Y)uXI = III IYI + > 111 = a(O). This contradiction yields the desired result. 0 A vertex v of a graph G is simplicial if every two vertices of N( v, 0) are adjacent in G. Equivalently, a simplicial vertex is a vertex that appears in exactly one clique. A clique of a graph G containing at least one simplicial vertex of G is called a simplex of G. A graph G is said to be simplicial if every vertex of 0 is simplicial or is adjacent to a simplicial vertex of O. Certainly, if G is simplicial and S1, S2, ... , Sn are the simplexes of G, then V(G) = Ui=l V(Si)' The next result is a special case of Proposition 2.2.