A note on hyperopic cops and robber

TL;DR

This paper investigates a variant of the Cops and Robber game where the robber is invisible unless outside the cops' common neighborhood, analyzing bounds on the hyperopic cop number and related graph parameters.

Contribution

It introduces the hyperopic cop number, relates it to a new graph parameter, and studies bounds for specific graph classes like diameter 2 graphs and graph products.

Findings

Bounds on hyperopic cop number for diameter 2 graphs

Relation between common neighborhood set size and hyperopic cop number

Analysis of hyperopic cop number in graph joins and Cartesian products

Abstract

We explore a variant of the game of Cops and Robber introduced by Bonato et al.~where the robber is invisible unless outside the common neighbourhood of the cops. The hyperopic cop number is analogous to the cop number and we investigate bounds on this quantity. We define a small common neighbourhood set and relate the minimum cardinality of this graph parameter to the hyperopic cop number. We consider diameter 2 graphs, particularly the join of two graphs, as well as Cartesian products.