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...

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.

Let D be a digraph. Its reverse digraph, D−1, is obtained by reversing all arcs of D. We show that the domination numbers of D and D−1 can be different if D is a Cayley digraph. The smallest groups admitting Cayley digraphs with this property are the alternating group A4 and the dihedral group D6, both on 12 elements. Then, for each n ≥ 6 we find a Cayley digraph D on the dihedral group Dn such...

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman k-dominating function (STRkDF) on D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−(v) f(x) ≥ k for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight o...

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The w...

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...