Directed Domination in Oriented Graphs

TL;DR

This paper explores the concept of directed domination in graphs, defining the directed domination number, and establishes bounds and exact values for specific graph classes, extending previous work on complete graphs.

Contribution

It introduces the directed domination number for all graphs, relates it to the independence number for bipartite graphs, and provides bounds and exact values.

Findings

For bipartite graphs, the directed domination number equals the independence number.

Several lower and upper bounds on the directed domination number are established.

The concept extends the study of domination from undirected to directed graphs.

Abstract

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $\gamma(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $\Gamma_d(G)$, which is the maximum directed domination number $\gamma(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. We extend this notion to directed domination of all graphs. If $\alpha$ denotes the independence number of a graph $G$, we show that if $G$ is a bipartite graph, we show that $\Gamma_d(G) = \alpha$. We present several lower and upper bounds on the directed domination number.