# Cops, robbers, and burning bridges

## TL;DR

This paper introduces the bridge-burning cop number, a new graph parameter in a variant of Cops and Robbers where edges are deleted after traversal, and provides exact values, algorithms, and bounds for various graph classes.

## Contribution

It defines the bridge-burning cop number, computes it for elementary graphs, develops a polynomial-time algorithm for trees, and analyzes capture times on grids, tori, and hypercubes.

## Key findings

- Exact $c_b(G)$ for several elementary graphs
- Polynomial-time algorithm for trees
- Bounds on capture time for specific graph classes

## Abstract

We consider a variant of Cops and Robbers wherein each edge traversed by the robber is deleted from the graph. The focus is on determining the minimum number of cops needed to capture a robber on a graph $G$, called the {\em bridge-burning cop number} of $G$ and denoted $c_b(G)$. We determine $c_b(G)$ exactly for several elementary classes of graphs and give a polynomial-time algorithm to compute $c_b(T)$ when $T$ is a tree. We also study two-dimensional square grids and tori, as well as hypercubes, and we give bounds on the capture time of a graph (the minimum number of rounds needed for a single cop to capture a robber on $G$, provided that $c_b(G) = 1$).