Total domination number of the conjunction

The total domination number of a graph is the size of a smallest total dominating set , where a total dominating set is a set of vertices of the graph such that all vertices including those in the set itself have a neighbor in the set. Total dominating numbers are defined only for graphs having no isolated vertex plus the trivial case of the singleton graph. For example, in the Petersen graph illustrated above, since the set is a minimum dominating set left figure , while since is a minimum total dominating set right figure. For any simple graph with no isolated points, the total domination number and ordinary domination number satisfy.

Skip to search form Skip to main content. The total outer-independent domination number of a graph G is the minimum cardinality of a total outer-independent dominating set of G. First we discuss the basic properties of total outerindependent domination in graphs. View PDF.

User options. Login Register Support. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.

Chinese Annals of Mathematics, Series B. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D , and the set V G D is independent. The 2-domination total outer-independent domination, respectively number of a graph G is the minimum cardinality of a 2-dominating total outer-independent dominating, respectively set of G.