The $k$-visibility Localization Game

We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the $k$-visibility localization number ${\zeta }_k$, where $k$ is a non-negative integer. We give bounds on $k$-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all $k$, we give a family of trees with unbounded ${\zeta }_k$ values. Extending results known for the localization number, we show that for $k\ge 2$, every tree contains a subdivision with ${\zeta }_k=1$. For many $n$, we give the exact value of ${\zeta }_k$ for the $n\times n$ Cartesian grid graphs, with the remaining cases being one of two values as long as $n$ is sufficiently large. These examples also illustrate that ${\zeta }_i\mathrm{\ne }{\zeta }_j$ for all distinct choices of $i$ and $j\mathrm{.}$