Orienting edges to fight fire in graphs

TL;DR

This paper introduces a new variant of the Firefighter Problem where firefighters can orient edges before a fire starts, providing bounds and strategies for saving vertices across various graph classes.

Contribution

It proposes and analyzes an oriented edge variant of the Firefighter Problem, offering optimal strategies and bounds for different graph classes and parameters.

Findings

Optimal firefighting strategies for complete and bipartite graphs.

Lower bounds based on graph parameters like chromatic number and treewidth.

Saving all but two vertices in subcubic graphs is optimal.

Abstract

We investigate a new oriented variant of the Firefighter Problem. In the traditional Firefighter Problem, a fire breaks out at a given vertex of a graph, and at each time interval spreads to neighbouring vertices that have not been protected, while a constant number of vertices are protected at each time interval. In the version of the problem considered here, the firefighters are able to orient the edges of the graph before the fire breaks out, but the fire could start at any vertex. We consider this problem when played on a graph in one of several graph classes, and give upper and lower bounds on the number of vertices that can be saved. In particular, when one firefighter is available at each time interval, and the given graph is a complete graph, or a complete bipartite graph, we present firefighting strategies that are provably optimal. We also provide lower bounds on the number of vertices that can be saved as a function of the chromatic number, of the maximum degree, and of the treewidth of a graph. For a subcubic graph, we show that the firefighters can save all but two vertices, and this is best possible.