Efficient total domination in digraphs

Efficient total domination in digraphs

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X . We study graphs that permit an orientation having such a set and give complexity results and characterizations concerning this question. Furthermore, we study the computational comp...

Efficient open domination in digraphs

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...

Total and connected domination in digraphs

Inverse Domination and Inverse Total Domination in Digraphs

I. Introduction In this paper, D=(V, A) is a finite, directed graph with neither loops nor multiple arcs (but pairs of opposite arcs are allowed) and G=(V, E) is a finite, undirected graph with neither loops nor multiple edges. For basic terminology, we refer to Chartrand and Lesniak [2]. A set S of vertices in a graph G=(V, E) is a dominating set if every vertex in V – S is adjacent to some ve...

On weighted efficient total domination

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominable graph (G is etd). In this paper, we prove NP-completeness of the etd decision problem on the class of planar bipartite graphs of maximum degree 3. Furthermore, we give an effic...