Catching a mouse on a tree

Abstract: In this paper we consider a pursuit-evasion game on a graph. A team of cats, which may choose any vertex of the graph at any turn, tries to catch an invisible mouse, which is constrained to moving along the vertices of the graph. Our main focus shall be on trees. We prove that $\lceil (1/2)\log_2(n)\rceil$ cats can always catch a mouse on a tree of order $n$ and give a collection of trees where the mouse can avoid being caught by $ (1/4 - o(1))\log_2(n)$ cats.