The Restrained Domination and Independent Restrained Domination in Extending Supergrid Graphs

Abstract

Let G be a graph with vertex set V(G) and edge set E(G). A set \(D\subseteq V(G)\) is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. A restrained dominating set of G is a dominating set S such that every vertex not in S is adjacent to another vertex in \(V(G)-S\). An independent restrained dominating set of G is a restrained dominating set such that it is also an independent set. The domination (resp., restrained domination, and independent restrained domination) number of G, denoted by \(\gamma (G)\) (resp., \(\gamma _{\mathrm {r}}(G)\), and \(\gamma _{\mathrm {ir}}(G)\)) is the minimum cardinality of a dominating (resp., a restrained dominating, and an independent restrained dominating) set of G. The domination (resp., restrained domination, and independent restrained domination) problem on a graph G is to compute a dominating (resp., a restrained dominating, and an independent restrained dominating) set of G with size \(\gamma (G)\) (resp., \(\gamma _{\mathrm {r}}(G)\), and \(\gamma _{\mathrm {ir}}(G)\)). Extending supergrid graphs are a natural extension of grid graphs. They are first appeared here and contain grid, supergrid, triangular supergrid, and diagonal supergrid graphs as subclasses. The domination problem on grid graphs was known to be NP-complete, and hence it is NP-complete for extending supergrid graphs. However, the complexities of the restrained and independent restrained domination problems on (extending) supergrid graphs are still unknown. In this paper, we will prove these two problems to be NP-complete for diagonal supergrid graphs, and hence they are NP-complete for extending supergrid graphs. These results can be easily applied to supergrid graphs. Then, we compute \(\gamma _{\mathrm {r}}(R_{m\times n})\) and \(\gamma _{\mathrm {ir}}(R_{m\times n})\), and verify that \(\gamma _{\mathrm {r}}(R_{m\times n}) = \gamma _{\mathrm {ir}}(R_{m\times n}) = \gamma (R_{m\times n})\) for rectangular supergrid graph \(R_{m\times n}\) which form a subclass of diagonal supergrid graphs excluding paths.