Testing Bipartiteness of Geometric Intersection Graphs

TL;DR

This paper presents efficient algorithms for testing bipartiteness of intersection graphs of various geometric objects, achieving near-linear time for certain classes, and explores complexity boundaries for related coloring problems.

Contribution

It introduces subquadratic algorithms for bipartiteness testing of intersection graphs of geometric objects, extending to connectivity and k-colorability problems.

Findings

Bipartiteness testing for line segments, polygons, and balls in R^d can be done in O(n log n) time.

Connectivity testing for unit balls in R^d is as complex as Euclidean MST construction.

Testing k-colorability for k>2 is NP-complete for line segments and planar disks.

Abstract

We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of balls in R^d, 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. For unit balls in R^d, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and hence is unlikely to be solved as efficiently as bipartiteness testing. For line segments or planar disks, testing k-colorability of intersection graphs for k>2 is NP-complete.