A graph with n vertices is well-covered if every maximal independent set is a maximum independent set, and very well-covered if every maximal independent set has size n ⁄2. In this paper, we study these graphs from an algorithmic complexity point of view. We show that wellcovered graph recognition is co-NP-complete and that several other problems are NP-complete for well-covered graphs. A numbe...

‎In this paper we give a characterization of unmixed tripartite‎ ‎graphs under certain conditions which is a generalization of a‎ ‎result of Villarreal on bipartite graphs‎. ‎For bipartite graphs two‎ ‎different characterizations were given by Ravindra and Villarreal‎. ‎We show that these two characterizations imply each other‎.

A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) (M. D. Plummer, 1970). 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 (I. Gutman and F. Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured th...

A graph G is fractional k-covered if for each edge e of G, there exists a fractional k-factor h, such that h(e) = 1. If k = 2, then a fractional k-covered graph is called a fractional 2-covered graph. The binding number bind(G) is defined as follows, bind(G) = min{ |NG(X)| |X| : Ø = X ⊆ V (G), NG(X) = V (G)}. In this paper, it is proved that G is fractional 2-covered if δ(G) ≥ 4 and bind(G) > 5...

We introduce the concept of combed graphs and present an ear decomposition theorem for this class of graphs. This theorem includes the well known ear decomposition theorem for matching covered graphs proved by Lovász and Plummer. Then we use the ear decomposition theorem to show that any two edges of a 2-connected combed graph lie in a balanced circuit of an equivalent combed graph. This result...

T. S. Michael and N. Traves (2002) provided examples of wellcovered graphs whose independence polynomials are not unimodal. A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain conditions, any well-covered graph equals G∗ for some G, where G∗ is the graph obtained from G by appending a single pendant edge to each vertex of G. Y. Alavi, P. J. Malde, A. J. Schwenk and P. E...

The stability number of a graph G, denoted by α(G), is the cardinality of a maximum stable set, and μ(G) is the cardinality of a maximum matching in G. If α(G) + μ(G) equals its order, then G is a König-Egerváry graph. We call G an α-square-stable graph if α(G) = α(G), where G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann, [18]. In this paper we o...