Trivially extendable graphs

Abstract

Let G be a simple graph. Let k be a positive integer. G is said to be k-extendable if every independent set of cardinality k is contained in a maximum independent set of G. G is said to be trivially extendable if G is not k-extendable for 1 ? k ? (?0(G) ? 1). A well covered graph is one in which every maximal independent set is maximum. Study of k-extendable graphs has been made in [7,8,9]. In this paper a study of trivially extendable graphs is made. Characterization of graphs with ?0(G) = (n ? 3) and which is trivially extendable has been done. Similarly graphs with ?0(G) = (n ? 2) is also studied for trivial extensibility.