We study graphs in which the maximum and the minimum sizes of a maximal independent set differ by exactly one. We call these graphs almost well-covered, in analogy with the class of well-covered graphs, in which all maximal independent sets have the same size. A characterization of graphs of girth at least 8 having exactly two different sizes of maximal independent sets due to Finbow, Hartnell,...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...

We present a new energy-minimization framework for the graph isomorphism problem which is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feat...

A graph is ptolemaic if and only if it is both chordal and distancehereditary. Thus, a ptolemaic graph G has two kinds of intersection graph representations: one from being chordal, and the other from being distance-hereditary. The first of these, called a clique tree representation, is easily generated from the clique graph of G (the intersection graph of the maximal complete subgraphs of G). ...