Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.Comment: 12 pages, 12 figure

A characterization of certain families of well-covered circulant graphs

viii, 202 leaves : ill. ; 29 cm.Includes abstract.Includes bibliographical references (leaves 200-202).A graph G is said to be well-covered if every maximal independent set is a maximum independent set. The concept of well-coveredness is of interest due to the fact that determining the independence number of an arbitrary graph is NP- complete, and yet for a well-covered graph it can be established simply by finding any one maximal independent set. A circulant graph C (n,S) is defined for a positive integer n and a subset S of the integers 1, 2, …, [n/2], called the connections. The vertex set is Z[subscript n], the integers modulo n. There is an edge joining two vertices j and i if and only if the difference |j-i| is in the set S. In this thesis, we investigate various families of circulant graphs. Though the recognition problem for well-covered circulant graphs is co-NP-complete, we are able to determine some general properties regarding these families and to obtain a characterization