Disjunctive Total Domination Numbers of Grid Graphs

For a graph G = (V, E), a subset D of V is a total dominating set of G if every vertex in V has to be adjacent to a vertex of D. Vertex subset D is a disjunctive total dominating set if every vertex of V is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The disjunctive total domination problem on G is to find a disjunctive total dominating set D of the minimum cardinality. The cardinality of a minimum disjunctive total dominating set of G is called the disjunctive total domination number of G. In this paper, we study the disjunctive total domination problem on grid graphs. We propose upper bounds for general grid graphs. In particular, some bounds are optimal for some special cases.