Domination Critical Semigraphs

Abstract

Sampathkumar [1] introduced a new type of generalization to graphs, called Semigraphs. A semigraph G = (V, X) on the set of vertices V and the set of edges X consists of n-tuples (u1,u2,un) of distinct elements belonging to the set V for various n ? 2, with the following conditions : (1) Any n-tuple (u1,u2,un) = (un,un-1,...,u1) and (2) Any two such tuples have at most one element in common. S. S. Kamath and R. S. Bhat [3] introduced domination in semigraphs. Two vertices u and v are said to a-dominate each other if they are adjacent. A set D ? V(G) is an adjacent dominating set (ad-set) if every vertex in V - D is adjacent to a vertex in D. The minimum cardinality of an ad-set D is called adjacency domination number of G and is denoted by ?a.. ?a(G) may increase or decrease by the removal of a vertex or an edge from G. A vertex v of a semigraph G is said to be ?a - critical if ?a(G - v) ? ?a(G); if ?a(G - v) = ?a(G), then v is ?a - redundatnt. The main objective of this paper is to study this phenomenon on the vertices and edges of a semigraph. 2005 Elsevier Ltd. All rights reserved.