In a finite undirected graph G = (V; E), a homogeneous set is a set U V of at least two vertices such that every vertex in V n U is either adjacent to all vertices of U or nonadjacent to all of them. A graph is prime if it does not have a homogeneous set. We investigate the minimum...

A set W ` V (G) is called homogeneous in a graph G if 2 jW j jV (G)j \Gamma 1, and N(x)nW = N(y)nW for each x; y 2 W . A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H , and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)nfHg. We denote by Ext(G) the set of all minimal extensions of a graph G. Zverovich [4]

Modular decomposition of graphs is a powerful tool for designing efficient algorithms for algorithmic graph problems such as the Maximum Weight Stable Set Problem and the Maximum Weight Clique Problem. Using this tool we obtain O(nm) time algorithms for the Maximum Weight Stable Set Problem on (chair, co-P)-free, (chair,P5)-free and (chair,bull)-free graphs. Moreover, our algorithms are robust in the sense that we do not have to check in advance whether the input graphs are indeed (chair, co-P)-free or (chair,P5)-free or (chair,bull)-free.

A set W ` V (G) is called homogeneous in a graph G if 2 jW j jV (G)j \Gamma 1, and N(x)nW = N(y)nW for each x; y 2 W . A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H , and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)nfHg. We denote by Ext(G) the set of all minimal extensions of a graph G.

We specify Reducing Copath Method of Zverovich [4] for some types of homogeneous sets.

We introduce a hereditary class of domination reducible graphs where the minimum dominating set problem is polynomially solvable, and characterize this class in terms of forbidden induced subgraphs. RRR 23-2001 Page 1 1

Let G and H be graphs. A substitution of H in G instead of a vertex v 2 V (G) is the graph G(v ! H), which consists of disjoint union of H and G \Gamma v with the additional edge-set fxy : x 2 V (H); y 2 NG (v)g. For a hereditary class of graphs P , the substitutional closure of P is defined as the class P consisting of all graphs which can be obtained from graphs in P by repeated substitutions. We characterize the substitutional closure (P 5 [ K 1 ; P 5 [ K 1 )-free graphs in terms of forbidden induced subgraphs. RRR 22-2001 Page 1 1

Let G and H be graphs. A substitution of H in G instead of a vertex v 2 V (G) is the graph G(v ! H), which consists of disjoint union of H and G \Gamma v with the additional edge-set fxy : x 2 V (H); y 2 NG (v)g. For a hereditary class of graphs P , the substitutional closure of P is defined as the class P consisting of all graphs which can be obtained from graphs in P by repeated substitutions. We give forbidden induced subgraph characterizations of the substitutional closure of Claw-free graphs, and (K 1 [ P 4 )-free graphs. Note that the weighted stability number problem can be solved in polynomial time for (Claw-free graphs) , and ((K 1 [ P 4 )-free graphs) . Also, these classes are polynomially recognizible.

The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classified by listing them as a set of explicit families.