The watchman's walk problem on directed graphs

TL;DR

This paper extends the watchman's walk problem to directed graphs, focusing on tournaments and complete multipartite graphs, providing bounds on the watchman number and exploring its relation to domination variants.

Contribution

It introduces the watchman's walk problem for directed graphs and establishes bounds for specific graph families, advancing understanding of domination in directed settings.

Findings

Bounds on the watchman number for tournaments.

Bounds on the watchman number for orientations of complete multipartite graphs.

Relationship between watchman number and domination variants.

Abstract

In a graph, a watchman's walk is a minimum closed dominating walk. Given a graph $G$ and a single watchman, the length of a watchman's walk in $G$ (the watchman number) is denoted by $w(G)$ and the typical goals of the watchman's walk problem is to determine $w(G)$ and find a watchman's walk in $G$. In this paper, we extend the watchman's walk problem to directed graphs. In a directed graph, we say that the watchman can only move to and see the vertices that are adjacent to him relative to outgoing arcs. That is, a watchman's walk is oriented and domination occurs in the direction of the arcs. The directed graphs this paper focuses on are families of tournaments and orientations of complete multipartite graphs. We give bounds on the watchman number and discuss its relationship to variants of the domination number.