Some Algorithmic Results on Restrained Domination in Graphs

A set $D\subseteq V$ 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 $V \setminus D$. The \textsc{Minimum Restrained Domination} problem is to find a restrained dominating set of minimum cardinality. Given a graph $G$, and a positive integer $k$, the \textsc{Restrained Domination Decision} problem is to decide whether $G$ has a restrained dominating set of cardinality a most $k$. The \textsc{Restrained Domination Decision} problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this NP-completeness result by showing that the \textsc{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 \textsc{Minimum Restrained Domination} problem in block graphs, a subclass of doubly chordal graphs. The \textsc{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 \textsc{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.