Regular Restrained Domination in Middle Graph

Abstract

In this paper, we introduce the new concept called regular restrained domination in middle graph.A set S ? V[M(G)] is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and another vertex in V-S. Note that every graph has a restrained dominating set, since S=V is such a set. Let ?rr[M(G)] denote the size of a smallest restrained dominating set. Also we study the graph theoretic properties of ?rr[M(G)] and many bounds were obtained in terms of elements of G and its relationships with other domination parameters were found.