Restrained Domination in Some Subclasses of Chordal 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. 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.