The Localization game on locally finite trees

Abstract

We study the Localization game on locally finite graphs and trees, where each vertex has finite degree. As in finite graphs, we prove that any locally finite graph contains a subdivision where one cop can capture the robber. In contrast to the finite case, for $n$ a positive integer, we construct a locally finite tree with localization number $n$ for any choice of $n$. Such trees contain uncountably many ends, and we show this is necessary by proving that graphs with countably many ends have localization number at most 2. We finish with questions on characterizing the localization number of locally finite trees.