Some Algorithmic Results on Restrained Domination in Graphs

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating set of cardinality a most k. The RESTRAINED DOMINATION DECISION problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this NPcompleteness result by showing that the RESTRAINED DOMINATION DECISION problem remains NP-complete for doubly chordal graphs, a subclass of chordal graphs. We also propose a polynomial time algorithm to solve the MINIMUM RESTRAINED DOMINATION problem in block graphs, a subclass of doubly chordal graphs. The RESTRAINED DOMINATION DECISION problem is also known to be NP-complete for split graphs. We propose a polynomial time algorithm to compute a minimum restrained dominating set of threshold graphs, a subclass of split graphs. In addition, we also propose polynomial time algorithms to solve the MINIMUM RESTRAINED DOMINATION problem in cographs and chain graphs. Finally, we give a new improved upper bound on the restrained domination number, cardinality of a minimum restrained dominating set in terms of number of vertices and minimum degree of graph. We also give a randomized algorithm to find a restrained dominating set whose cardinality satisfy our upper bound with a positive probability.