The Paired Restrained-Domination Problem in Supergrid Graphs

Let G be a graph with vertex set V(G) and edge set E(G). A set ⊆ V(G) is a dominating set of G if every vertex not in D is adjacent to one vertex in D. The domination problem on G is to compute a dominating set of G with minimum cardinality. A set S ⊆ V(G) is called a paired restrained-dominating set of G if S is a dominating set of G, the subgraph induced by S contains a perfect matching, and the subgraph induced by V(G)−S contains no isolated vertex. The paired restrained- domination number γ
p
(G) of a graph G is the minimum size of a paired restrained-dominating set in G. The paired restrained- domination problem on a graph G is to find a paired restrained- dominating set of G with cardinality γ
p
(G), and is first introduced here. Extending supergrid graphs form a superclass of grid graphs, diagonal supergrid graphs, and supergrid graphs. The domination problem for grid graphs was known to be NP-complete and hence it is NP-complete on extending supergrid graphs. In the past, we have proved the domination problem on supergrid graphs to be NP-complete. The complexity of the paired restrained-domination problem on grid, diagonal supergrid, and supergrid graphs is still unknown. In this paper, we will prove it to be NP-complete for diagonal supergrid graphs, and hence it is NP-complete for extending supergrid graphs. This result can be extended to supergrid graphs. We then provide an upper bound of γ
p
(R
m×n
) for rectangular supergrid graph R
m×n
.