Restrained domination in graphs Original Research Article

In this paper, we initiate the study of a variation of standard domination, namely restrained domination. Let G=(V,E) be a graph. A restrained dominating set is a set S⊆V where every vertex in V−S is adjacent to a vertex in S as well as another vertex in V−S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. We determine best possible upper and lower bounds for γr(G), characterize those graphs achieving these bounds and find best possible upper and lower bounds for γr(G)+γr(G) where G is a connected graph. Finally, we give a linear algorithm for determining γr(T) for any tree and show that the decision problem for γr(G) is NP-complete even for bipartite and chordal graphs.