Restrained domination in graphs

Abstract 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 γ r ( G ), is the smallest cardinality of a restrained dominating set of G . We determine best possible upper and lower bounds for γ r ( G ), characterize those graphs achieving these bounds and find best possible upper and lower bounds for γ r (G)+γ r ( G ) where G is a connected graph. Finally, we give a linear algorithm for determining γ r ( T ) for any tree and show that the decision problem for γ r ( G ) is NP-complete even for bipartite and chordal graphs.