# An evasion game on a graph

## TL;DR

This paper studies a pursuit and evasion game on graphs, characterizing the graphs where the evader can always be caught and determining optimal capture times.

## Contribution

It introduces a new pursuit-evasion game on graphs and characterizes the winning graphs as certain trees without a specific subgraph, providing optimal capture times.

## Key findings

- Graphs where the evader can guarantee capture are exactly certain trees without a forbidden subgraph.
- The paper determines the best possible capture times on these graphs.
- It establishes a precise graph-theoretic characterization of winning conditions.

## Abstract

This paper introduced a pursuit and evasion game to be played on a connected graph. One player moves invisibly around the graph, and the other player must guess his position. At each time step the second player guesses a vertex, winning if it is the current location of the first player; if not the first player must move along an edge. It is shown that the graphs on which the second player can guarantee to win are precisely the trees that do not contain a particular forbidden subgraph, and best possible capture times on such graphs are obtained.