Split Middle Domination in Graphs

Abstract

The middle graph of a graph G, denoted by M(G) is a graph whose vertex set is V(G)?E(G) and two vertices are adjacent if they are adjacent edges of G or one is a vertex and other is an edge incident with it. A dominating set D of M(G) is called split dominating set of M(G) if the induced subgraph ?V[M(G) ]-D? is disconnected. The minimum cardinality of D is called the split middle domination number of G and is denoted by? ??_SM (G). In this paper many bound on ? ??_SM (G) were obtained in terms of the vertices, edges and many other different parameters of G but not in terms of the elements of M(G). Further its relation with other different parameters are also developed.