An upper bound for the restrained domination number of a graph with minimum degree at least two in terms of order and minimum degree

Abstract

AbstractLet G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γr(G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is a connected graph of order n and minimum degree δ and not isomorphic to one of nine exceptional graphs, then γr(G)≤n−δ+12