Let G be a digraph. A set S ⊆ V (G) is called an efficient total dominating set if the set of open out-neighborhoods N−(v) ∈ S is a partition of V (G). We say that G is efficiently open-dominated if both G and its reverse digraph G− have an efficient total dominating set. Some properties of efficiently open dominated digraphs are presented. Special attention is given to tournaments and directed...

We show that the total domination number of a simple connected graph is greater than the average distance of the graph minus one-half, and that this inequality is best possible. In addition, we show that the domination number of the graph is greater than two-thirds of the average distance minus one-third, and that this inequality is best possible. Although the latter inequality is a corollary t...

Distance-hereditary graphs are in important graph class theory, as they well-placed the hierarchy and permit many algorithmic results. We investigate structural advantages of a directed version this well-researched class. Since previously defined distance-hereditary digraphs do not recursive structure, we define twin-distance-hereditary graphs, which can be constructed by several twin pendant v...

A domination graph of a digraph D, dom (D), is created using thc vertex set of D and edge uv E E (dom (D)) whenever (u, z) E A (D) or (v, z) E A (D) for any other vertex z E V (D). Here, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. Specifically, digraphs are completely characterized where UG (D) is the union of two disjoint paths.

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.