Cops and Robbers on Intersection Graphs [article]

Tomáš Gavenčiak, Przemysław Gordinowicz, Vít Jelínek, Pavel Klavík, Jan Kratochvíl
2016 arXiv   pre-print
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,
more » ... 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.
arXiv:1607.08058v1 fatcat:yyeeioogmbde5hjqtd2tb7proy