Hyperopic Cops and Robbers

TL;DR

This paper introduces the hyperopic cop number, a new variant of the Cops and Robbers game on graphs where the robber's visibility is limited, and explores its properties, bounds, and specific cases.

Contribution

It defines the hyperopic cop number, characterizes cop-win graphs, and analyzes bounds and properties for various graph classes, including planar and countable graphs.

Findings

Hyperopic cop number is at most 3 for planar graphs.

Characterization of cop-win graphs for the hyperopic variant.

Hyperopic cop density can vary in [0, 1/2] for chains of graphs.

Abstract

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in $[0,1/2].$