The Localization Game on Directed Graphs

TL;DR

This paper extends the localization game to directed graphs, providing bounds on the localization number, analyzing specific graph families, and exploring properties of random and regular tournaments.

Contribution

It introduces the localization number for directed graphs and establishes bounds, including tight bounds via strong components and linear programming, advancing understanding of the game on digraphs.

Findings

Bound on localization number via strong components

Linear programming bounds for directed graphs

Localization number of random and regular tournaments

Abstract

In the Localization game played on graphs, a set of cops uses distance probes to identify the location of an invisible robber. We present an extension of the game and its main parameter, the localization number, to directed graphs. We present several bounds on the localization number of a directed graphs, including a tight bound via strong components, a bound using a linear programming problem on hypergraphs, and bounds in terms of pathwidth and DAG-width. A family of digraphs of order $n$ is given with localization number $(1-o(1))n/2$. We investigate the localization number of random and quasi-random tournaments, and apply our results to doubly regular tournaments, which include Paley tournaments.