Packing and domination parameters in digraphs

Abstract

Given a digraph D = (V, A), a set B ⊂ V is a packing set in D if there are no arcs joining vertices of B and for any two vertices x, y ∈ B the sets of in-neighbors of x and y are disjoint. The set S is a dominating set (an open dominating set) in D if every vertex not in S (in V ) has an in-neighbor in S. Moreover, a dominating set S is called a total dominating set if the subgraph induced by S has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We prove that the two optimal sets concerning packing and domination achieve the same value for directed trees, and give some applications of it. We also show analogous equalities for all connected contrafunctional digraphs, and characterize all such digraphs D for which such equalities are satisfied. Moreover, sharp bounds on the maximum and the minimum cardinalities of packing and dominating sets, respectively, are given for digraphs. Finally, we present solutions for two open problems, concerning total and open dominating sets of minimum cardinality, pointed out in [Australas. J. Combin. 39 (2007), 283-292].

Domination in digraphs and their products

“…Theorem 2 extends the recent result of Mojdeh, Samadi and González Yero [17,Theorem 5] who proved it for oriented trees. (An oriented tree is a digraph obtained from an undirected tree by orienting each of its edges in one direction.…”

Total 2-domination number in digraphs and its dual parameter

“…Mojdeh et al [21] proved that ρ(T ) = γ(T ), for all directed tree T . Although such a result does not hold for L t 2 (T ) and γ t 2 (T ), one can ask for the family of all directed trees for which these two parameters are the same.…”

“…However, many domination parameters have been investigated in digraphs. For example, the reader can consult the papers [5,15,19], and [6,7,21] in recent years. This paper also contributes to decreasing the unbalanced situation.…”

Double domination and total $2$-domination in digraphs and their dual problems

“…A subset S ⊆ V (D) (S ⊆ V (G)) is a dominating set in D (G) if all vertices are dominated by the vertices in S. The domination number γ(D) (γ(G)) is the minimum cardinality of a dominating set in the digraph D (graph G). The concept of domination in digraphs was introduced by Fu [10] and has been extensively studied in several papers including [6,14,20,21], while the very well-known same topic in graphs was intdoduced by Berge [5] and Ore [23]. The reader is referred to [19] for more details on this topic.…”

“…Thus, at least one of the vertices x 1 , • • • , x q does not belong to B ∩ B −1 . ( ) Mojdeh et al [21] proved that ρ(T ) = γ(T ), for all directed tree T . Although such a result does not hold for L t 2 (T ) and γ t ×2 (T ), one can ask for the family of all directed trees for which these two parameters are the same.…”