Paired restraint domination in extended supergrid graphs

Consider a graph G with vertex set V(G) and edge set E(G). A subset D of V(G) is said to be a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. If, in addition, every vertex not in a dominating set R of G is adjacent to at least one vertex in V(G)-R, then R is called a restrained dominating set of G. A paired restraint dominating set S of a graph G is a restrained dominating set of G satisfying that the induced subgraph by S contains a perfect matching. The problems of computing the dominating set, restrained dominating set, and paired restraint dominating set with minimum cardinality are referred to as the domination, restrained domination, and paired restraint domination problems, respectively. The paired restraint domination problem and its applications are first proposed here. Our focus is on examining the complexity of the proposed problem on extended supergrid graphs and their subclasses, which include grid, diagonal supergrid, and (original) supergrid graphs. The domination problem is known to be NP-complete on grid graphs and, therefore, also on extended supergrid graphs. Our previous research demonstrated that the domination and restrained domination problems on diagonal and original supergrid graphs are NP-complete. However, the complexity of the paired restraint domination problem on grid, diagonal supergrid, and original supergrid graphs remains unknown. The NP-completeness of the paired restraint domination problem on diagonal supergrid graphs is demonstrated in this paper, and this finding is also applicable to the original supergrid graphs and planar graphs with maximum degree 4. We then examine a subclass of diagonal and original supergrid graphs known as rectangular supergrid graphs. These graphs are distinguished by a rectangular shape consisting of m rows and n columns of vertices. Specifically, we address the paired restraint domination problem on R-mxn and develop a linear time algorithm for 3 >= m >= 1 and n >= m . Then, when n >= m >= 4 , we obtain a tight upper bound on the minimum size of the paired restraint dominating set of R-mxn and then use this upper bound to establish one lower bound.