Packing and domination parameters in digraphs

TL;DR

This paper investigates the relationships and bounds of packing and domination parameters in directed graphs, establishing equalities for specific classes and solving open problems in the field.

Contribution

It characterizes when maximum packing and minimum dominating sets coincide in digraphs, especially in trees and contrafunctional digraphs, and provides bounds and solutions for open problems.

Findings

Maximum packing and minimum dominating sets are equal in directed trees.

Analogous equalities hold for connected contrafunctional digraphs.

Provides bounds and solutions for open problems in domination parameters.

Abstract

Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ 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].