Strong and Weak Domination, Full Sets and Domination Balance in Semigraphs

Abstract

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 ( u 1 , u 2 ,…, u n ) of distinct elements belonging to the set V for various n ≥ 2, with the following conditions : (1) Any n -tuple ( u 1 , U 2 ,…, u n ) = ( u n , u n -1 , …, u 1 ) 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 γ a. A vertex u strongly (weakly) a-dominates a vertex υ if, deg a u ≥ deg a υ (deg a u ≤ deg a υ) where deg a u is the number of vertices adjacent to u. A set D ⊆ V(G) is a strong (weak) adset [sad-set (wad-set)], if every vertex in V - D is strongly (weakly) a-dominated by at least one vertex in D. This paper presents some new results on strong (weak) domination in semigraphs.