On Evasion Games on Graphs

We consider an evasion game on a connected simple graph. We first show that the pursuit number of a graph G, the smallest k such that k pursuers win the game, is bounded above by the pathwidth of G. We next show that the pursuit number of G is two if and only if the pathwidth of G is one. We also show that for any integer \({w}\ge 2\), there exists a tree T such that the pursuit number of T is three and the pathwidth of T is w.