Restrained domination and its variants in extended supergrid graphs

Abstract

Domination problem is a well-known NP-complete problem for general graphs. In this paper, we will study its three variants, including restrained, independent restrained, and restrained-step dominations. A restrained dominating set of a graph is a set R of vertices of the graph such that every vertex of the graph is in R or adjacent to at least a vertex of R, and the subgraph induced by vertices not in R contains no isolated vertex. An independent restrained dominating set of a graph is a restrained dominating set such that its any two vertices are not adjacent. A restrained-step dominating set of a graph is a restrained dominating set such that the subgraph induced by vertices not in this set contains a perfect matching. Restrained domination, independent restrained domination, and restrained-step domination problems are to find a restrained dominating set, independent restrained dominating set, and restrained-step dominating set with the least number of vertices on a graph, respectively. In 2015, we define supergrid graphs for computing the tracks of computer embroidery sewing machine and 3D printer. In this paper, we expand it to the class of extended supergrid graphs. Extended supergrid graphs are a natural extension of supergrid graphs, and they contain grid, triangular grid, and supergrid graphs as subclasses. The complexities of the restrained related domination problems on extended supergrid graphs remain unknown. In this paper, we will prove the restrained and independent restrained domination problems on supergrid graphs is NP-complete. Using the same technique, the restrained domination problem on grid graphs can also be shown to be NP-complete. We then solve the restrained related domination problems on rectangular supergrid graphs, which form a subclass of supergrid graphs, in linear time. Finally, we provide a tight upper bound on the minimum size of (independent) restrained dominating set for rectangular triangular-supergrid graphs forming a subclass of triangular grid graphs.