The one-visibility Localization game

TL;DR

This paper introduces the one-visibility localization game, analyzing the new parameter ta_1, providing bounds for various graph classes, and exploring its properties and implications in graph theory.

Contribution

The paper defines the one-visibility localization number ta_1, establishes bounds for different graph classes, and relates it to existing graph parameters, advancing understanding of visibility-based pursuit games.

Findings

ta_1 is unbounded on k-ary trees.

ta_1 has an O(rac{}{2}) bound on K_h-minor free graphs.

Cartesian grids' ta_1 determined up to four values.

Abstract

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 $\zeta_1$. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for $\zeta_1$ on $k$-ary trees with $k\ge 2$ that differ by a multiplicative constant, showing that the parameter is unbounded on $k$-ary trees. We provide a $O(\sqrt{n})$ bound for $K_h$-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 $\zeta_1$ using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.