A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O (

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a recurrence relation, which shows that both graph polynomials are substitution instances of each other. We give some properties of the covered components polynomial an...

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove...

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove...

Abstract We prove that, for every graph F with at least one edge, there is a constant $$c_F$$ c F such that are graphs of arbitrarily large chromatic number and the same clique as in which -free induced subgraph has most . This generalises recent theorems Briański, Davi...

A snake in a graph G is defined to be a closed path in G without proper chords. Let Kd n be the product of d copies of the complete graph Kn. Wojciechowski [13] proved that for any d ≥ 2 the hypercube Kd 2 can be vertex covered with at most 16 disjoint snakes. Alsardary [6] proved that for any odd integer n ≥ 3,d ≥ 2 the graph Kd n can be vertex covered with 2n 3 snakes. We show that for any ev...

A graph G is a Zm-well-covered graph if |I1| ≡ |I2| (modm) for all maximal independent sets I1 and I2 in V (G) [3]. The recognition problem of Zm-well-covered graphs is a Co-NP-Complete problem. We give a characterization of Zm-well-covered graphs for chordal, simplicial and circular arc graphs. c © 2001 Elsevier Science B.V. All rights reserved.

If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M...

The notion of removable classes arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. The last (single or double) ear of an ear decomposition is defined as a removable class. Every matching covered graph not induced by a circuit has at least three removable classes. In this paper, we characterize matching covered graphs with precisely three re...

A graph is called perfect matching compact (briefly, PM -compact), if its perfect matching graph is complete. Matching-covered PM -compact bipartite graphs have been characterized. In this paper, we show that any PM -compact bipartite graph G with δ(G) ≥ 2 has an ear decomposition such that each graph in the decomposition sequence is also PM -compact, which implies that G is matching-covered.