Testing bipartiteness of geometric intersection graphs

Abstract

We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of an intersection graph of balls in d -dimensional Euclidean space, in time O ( n log n ). More generally, we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric objects. Our algorithms for these problems return either a bipartition of the input or an odd cycle in its intersection graph. We also consider lower bounds for connectivity and k -colorability problems of geometric intersection graphs. For unit balls in d dimensions, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and for line segments in the plane connectivity testing has the same lower bounds as Hopcroft's point-line incidence testing problem; therefore, for these problems, connectivity is unlikely to be solved as efficiently as bipartiteness. For line segments or planar disks, testing k -colorability of intersection graphs for k > 2 is NP-complete.