# An application of the Gy\'{a}rf\'{a}s path argument

## TL;DR

This paper adapts the Gyárfás path argument to establish bounds on the number of cops needed to capture a robber in a graph excluding a specific induced path, advancing understanding in graph pursuit games.

## Contribution

It introduces a novel adaptation of the Gyárfás path argument to analyze cop and robber game strategies in certain graph classes.

## Key findings

- Proves that $t-2$ cops suffice to capture the robber in at most $t-1$ moves.
- Establishes bounds for cop numbers in graphs excluding a $t$-vertex induced path.
- Provides a new technique for analyzing pursuit-evasion games in graph theory.

## Abstract

We adapt the Gy\'{a}rf\'{a}s path argument to prove that $t-2$ cops can capture a robber, in at most $t-1$ moves, in the game of cops and robbers played in a graph that does not contain the $t$-vertex path as an induced subgraph.