The one-visibility localization game

We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number ζ1. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for ζ1 on k-ary trees with k≥2 that differ by a multiplicative constant, showing that the parameter is unbounded on k-ary trees. We provide a O(n) bound for Kh-minor free graphs of order n, and we show Cartesian grids meet this bound by determining their one-visibility localization number up to four values. We present upper bounds on ζ1 using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.