Periodic Patrols on the Line and Other Networks

TL;DR

This paper analyzes periodic patrolling strategies on graphs, especially line graphs, for defending nodes against attacks of two periods, providing solutions based on graph fractional parameters and extending previous work to periodic scenarios.

Contribution

It extends the analysis of periodic patrolling games to arbitrary graphs and line graphs, offering solutions based on fractional graph parameters and modeling border patrol scenarios.

Findings

Game solvable using fractional covering and independence numbers for even periods

Complete solution for line graphs of any size

Models border patrolling problems in operational research

Abstract

We consider a patrolling game on a graph recently introduced by Alpern et al. (2011) where the Patroller wins if he is at the attacked node while the attack is taking place. This paper studies the periodic patrolling game in the case that the attack duration is two periods. We show that if the Patroller's period is even, the game can be solved on any graph by finding the fractional covering number and fractional independence number of the graph. We also give a complete solution to the periodic patrolling game on line graphs of arbitrary size, extending the work of Papadaki et al. (2016) to the periodic domain. This models the patrolling problem on a border or channel, which is related to a classical problem of operational research going back to Morse and Kimball (1951). A periodic patrol is required to start and end at the same location, for example the place where the Patroller leaves his car to begin a foot patrol.