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.

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality . Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1, 2]-factor FG, i.e. a spa...

Let G be a k-connected graph of order n. In [1], Bondy considered a degree sum condition for a graph to have a hamiltonian cycle, say, to be covered by one cycle. He proved that if σk+1(G) > (k+1)(n− 1)/2, then G has a hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. [4] proved that if k = 1 and σ3(G) ≥ n, then G can...

We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of Kn × G is found, provided that G has a largest greedy independent decomposition of length c < n. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, an...

Combining the concept of a fractional ( g , f )-covered graph with that ID-( )-factor-critical graph, we define covered graph. This paper reveals relationship between some parameters and existence graphs. A sufficient condition for being is presented. In addition, demonstrate sharpness main result in this by constructing special class. Furthermore, other parameters(such as binding number, tough...

A graph is well-covered if all its maximal independent sets are of the same cardinality. Deciding whether a given graph is well-covered is known to be NP-hard in general, and solvable in polynomial time, if the input is restricted to certain families of graphs. We present here a simple structural characterization of well-covered graphs and then apply it to the recognition problem. Apparently, p...

A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. A matching covered graph G is near-bipartite if it is nonbipartite and there is a removable double ear R such that G−R is matching covered and bipartite. In 2000, C. Little and I. Fischer characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [3]. However, their charac...