On Strong Split Middle Domination of a Graph

Abstract

The middle graph M(G) of graph G is obtained by inserting a vertex xi in the “middle” of each edge ei, 1 ? i ? |E(G)|, and adding the edge xixj for 1?i ? j ? |E(G)| if and only if ei and ej have a common vertex. A dominating set D of graph G is said to be a strong split dominating set of G if ?V(G) – D? is totally disconnected with at least two vertices. Strong split domination number is the minimum cardinality taken over all strong split dominating sets of G. In this paper we initiate the study of strong split middle domination of a graph. The strong split middle domination number of a graph G, denoted as ?ssm (G) is the minimum cardinality of strong split dominating set of M(G). In this paper many bounds on ?ssm(G) are obtained in terms of other domination parameters and elements of graph G. Also some equalities for ?ssm(G) are established.