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In this paper, 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 neighborhood 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 neighborhood 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. [ABSTRACT FROM AUTHOR]

Abstract: This paper establishes a necessary and sufficient condition for a CO-irredundant set of vertices of a graph to be maximal and shows that the smallest cardinality of a maximal CO-irredundant set in an n vertex graph with maximum degree is bounded below by for , for and for . This result is best possible and extremal graphs are characterised for . [Copyright &y& Elsevier]

Abstract: A necessary and sufficient condition for an open irredundant set of vertices of a graph to be maximal is obtained. This result is used to show that the smallest cardinality amongst the maximal open irredundant sets in an -vertex isolate-free graph with maximum degree is at least for , for and for . The bounds are the best possible. [Copyright &y& Elsevier]

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 ]. [ABSTRACT FROM AUTHOR]