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 ζk​, 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 ζk​ values. Extending results known
for the localization number, we show that for k≥2, every tree contains a
subdivision with ζk​=1. For many n, we give the exact value of
ζk​ for the n×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 ζiâ€‹î€ =ζj​ for all distinct choices of i and
$j.