# Capture-time Extremal Cop-Win Graphs

## TL;DR

This paper explores graphs where a single cop can catch a robber in the game of Cops and Robbers, focusing on those with the maximum possible capture time for their size, and introduces a new vertex-ranking approach.

## Contribution

It provides a new characterization of extremal cop-win graphs using a vertex rank framework and advances understanding of their structural properties.

## Key findings

- Partial classification of feasible rank configurations
- New bounds on capture time for extremal graphs
- Open questions on full classification of extremal graphs

## Abstract

We investigate extremal graphs related to the game of Cops and Robbers. We focus on graphs where a single cop can catch the robber; such graphs are called cop-win. The capture time of a cop-win graph is the minimum number of moves the cop needs to capture the robber. We consider graphs that are extremal with respect to capture time, i.e. their capture time is as large as possible given their order. We give a new characterization of the set of extremal graphs. For our alternative approach we assign a rank to each vertex of a graph, and then study which configurations of ranks are possible. We partially determine which configurations are possible, enough to prove some further extremal results. We leave a full classification as an open question.