Results on Total Domination and Total Restrained Domination in Grid Graphs

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. A set S of vertices in a graph G(V,E) is called a total restrained dominating set if every vertex v ∈ V is adjacent to an element of S and every vertex of V − S is adjacent to a vertex in V − S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Respectively the total restrained domination number of a graph G denoted by γtr(G) is the minimum cardinality of a total restrained dominating set in G. Here we investigate the problem of total domination numbers and total restrained domination numbers of some grid graphs (cartesian products of two paths Pn and Pm). And we determine the total domination numbers of Pn,n, P2n,2n+2, P2n,4n−1, and P2n,m for each n and m ≡ 2n (mod 2n + 1). Also we determine the total domination numbers of P8,n. We then show that for these grid graphs the total restrained domination number is equal to the total domination number.