The balanced connected subgraph problem for geometric intersection graphs

Abstract

We study the ▪ (shortly, ▪) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. We are given a ▪ graph G=(V,E) and an integer k. Each vertex in V is colored with either “▪” or “▪”. The BCS problem seeks an induced connected subgraph H of size at least k in G such that H is ▪, i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On the one hand, we prove that the BCS problem is NP-complete on the unit-disk, outer-string, grid, and unit-square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on the interval, circular-arc, and permutation graphs. Our algorithm for interval and circular-arc graphs solves the more general problem of computing a minimum cardinality ▪ in the same classes of graphs, which may be of independent interest.