Cops and Robbers on intersection graphs

Abstract

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs:• The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g . For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

On Strict (Outer-)Confluent Graphs

“…While this work provides the first in-depth study of SC and SOC graphs, a number of interesting open questions remain. One such question is motivated by our results on the cop-number of SOC graphs: we showed that SOC graphs are incomparable to most classes identified to have cop number two by Gavenciak et al [19], but we could not show such a result for the class of interval-filament graphs [21]. It seems likely that SOC graphs are contained in this class.…”

“…The cop number cop(G) of a graph G is the smallest integer k such that the cop-player has a winning strategy using k cop tokens. Gavenciak et al [19] showed that the cop number of outer-string graphs is between three and four, while the cop-number of many other interesting classes of intersection graphs, such as circle graphs and interval filament graphs, is two. We show that the cop number of SOC graphs is two as well.…”

“…In Section 5, we consider the so-called strict bipartite-outerconfluent drawings: by following up on an earlier result of Hui et al [32], we show that graphs which admit such a drawing are precisely the domino-free bipartite permutation graphs. Inspired by earlier work of Gavenčiak et al [19], we examine in Section 6 the cop number of SOC graphs and show that it is at most two. In Appendix E we show that many natural subclasses of outer-string graphs are incomparable to SOC graphs (see red, dashed boxes in Fig.…”

Greedy strategies and larger islands of tractability for conjunctive queries and constraint satisfaction problems

“…For general graphs on n vertices it is known that Ω( √ n) cops may be needed, and the celebrated Meyniel's conjecture [25] states that the cop number of a connected vertex graph is O( √ n). Moreover, exact or approximate values of the cop number of several classes of graphs have been derived so far, including plan graphs [7], bounded genus graphs [55], and intersection graphs [27], just to name a few.…”

A Cop and Drunken Robber Game on n-Dimensional Infinite-Grid Graphs

“…For example, Aigner and Fromme [12] established that if G is a planar graph, then c(G) = 3. This was a classical study which has been generalized in many topological graphs such as [13,14]. For other examples of CR studies, the readers may look at [15][16][17][18][19][20].…”