Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber’s turn. In the first round, the cops place themselves on a subset of vertices, after which the robber chooses a vertex to place himself. From the next round onwards, in the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. The cops win if they can capture the robber within a finite number of rounds; else the robber wins. A natural question in this game concerns the cop-number of a graph - the minimum number of cops needed to capture a robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, it was shown recently that the class of 1-planar graphs - graphs that can be drawn on the plane with at most one crossing per edge - does not have bounded cop-number. This paper initiates an investigation into how the distance between crossing pairs of edges influences a graph’s cop number. In particular, we look at Distance d Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance d of the robber. Let c_d(G) denote the minimum number of cops required in the graph G to capture a robber within distance d. We look at various classes of graphs, such as 1-plane graphs, k-plane graphs (graphs where each edge is crossed at most k times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then c_d(G) is bounded, for small values of d. For example, we show that if a graph G admits a drawing in which every pair of crossing edges is contained in a path of length at most 3, then c₄(G) ≤ 21. And if the drawing permits a stronger assumption that the endpoints of every crossing induce the complete graph K₄, then c₃(G) ≤ 9. The tools and techniques that we develop in this paper are sufficiently general, enabling us to examine graphs drawn not only on the sphere but also on orientable and non-orientable surfaces.

Cops and Robbers is a type of pursuit-evasion game played on a graph where a set of cops try to capture a single robber. The cops first choose their initial vertex positions, and later the robber chooses a vertex. The cops and robbers make their moves in alternate turns: in the cops' turn, every cop can either choose to move to an adjacent vertex or stay on the same vertex, and likewise the robber in his turn. If the cops can capture the robber in a finite number of rounds, the cops win, otherwise the robber wins. The cop-number of a graph is the minimum number of cops required to catch a robber in the graph. It has long been known that graphs embedded on surfaces (such as planar graphs and toroidal graphs) have a small cop-number. Recently, Durocher et al. [Graph Drawing, 2023] investigated the problem of cop-number for the class of 1-planar graphs, which are graphs that can be embedded in the plane such that each edge is crossed at most once. They showed that unlike planar graphs which require just three cops, 1-planar graphs have an unbounded cop-number. On the positive side, they showed that maximal 1-planar graphs require only three cops by crucially using the fact that the endpoints of every crossing in an embedded maximal 1-planar graph induce a K 4 . In this paper, we show that the cop-number remains bounded even under the relaxed condition that the endpoints induce at least three edges. More precisely, let an ×-crossing of an embedded 1-planar graph be a crossing whose endpoints induce a matching; i.e., there is no edge connecting the endpoints apart from the crossing edges themselves. We show that any 1-planar graph that can be embedded without ×-crossings has cop-number at most 21. Moreover, any 1-planar graph that can be embedded with at most γ ×-crossings has cop-number at most γ + 21 .

The problem of computing vertex and edge connectivity of a graph are classical problems in algorithmic graph theory. The focus of this paper is on computing these parameters for graphs drawn on the plane. A typical example of such graphs are planar graphs which can be embedded without any crossings. It has long been known that vertex and edge connectivity of planar embedded graphs can be computed in linear time. Very recently, Biedl and Murali extended the techniques from planar graphs to 1-plane graphs without ×-crossings, i.e., crossings whose endpoints induce a matching. While the tools used were novel, they were highly tailored to 1-plane graphs, and do not provide much leeway for further extension. In this paper, we develop alternate techniques that are simpler, have wider applications to near-planar graphs, and can be used to test both vertex and edge connectivity. Our technique works for all those embedded graphs where any pair of crossing edges are connected by a path that, roughly speaking, can be covered with few cells of the drawing. Important examples of such graphs include optimal 2-planar and optimal 3-planar graphs, d-map graphs, d-framed graphs, graphs with bounded crossing number, and k-plane graphs with bounded number of ×-crossings.

A graph is called 1-plane if it has an embedding in the plane where each edge is crossed at most once by another edge. A crossing of a 1-plane graph is called an ×-crossing if there are no other edges connecting the endpoints of the crossing (apart from the crossing pair of edges). In this paper, we show how to compute the vertex connectivity of a 1-plane graph G without ×-crossings in linear time. To do so, we show that for any two vertices u,v in a minimum separating set S, the distance between u and v in an auxiliary graph Λ(G) (obtained by planarizing G and then inserting into each face a new vertex adjacent to all vertices of the face) is small. It hence suffices to search for a minimum separating set in various subgraphs Λ_i of Λ(G) with small diameter. Since Λ(G) is planar, the subgraphs Λ_i have small treewidth. Each minimum separating set S then gives rise to a partition of Λ_i into three vertex sets with special properties; such a partition can be found via Courcelle’s theorem in linear time.