Restrained domination in some subclasses of chordal graphs

Abstract

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. The decision version of the MINIMUM RESTRAINED DOMINATION problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this NP-completeness result by showing that the 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.