Domination and Fractional 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...

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...

Fractional domination in prisms

Mynhardt has conjectured that if G is a graph such that γ(G) = γ(πG) for all generalized prisms πG then G is edgeless. The fractional analogue of this conjecture is established and proved by showing that, if G is a graph with edges, then γf (G×K2) > γf (G).

Stratification and Domination in Graphs and Digraphs