Balanced Connected Subgraph Problem in Geometric Intersection Graphs

TL;DR

This paper investigates the computational complexity of the Balanced Connected Subgraph problem in various geometric intersection graphs, providing both NP-hardness proofs and polynomial-time algorithms for specific graph classes.

Contribution

It establishes NP-hardness of BCS on several geometric graphs and offers polynomial algorithms for interval, circular-arc, and permutation graphs, along with an FPT algorithm for general graphs.

Findings

NP-hardness of BCS on unit disk, outer-string, grid, and square graphs.

Polynomial algorithms for BCS on interval, circular-arc, and permutation graphs.

FPT algorithm for BCS on general graphs.

Abstract

We study the Balanced Connected Subgraph(shortly, BCS) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. Given a vertex-colored graph $G=(V,E)$, where each vertex in $V$ is colored with either ``red'' or ``blue'', the BCS problem seeks a maximum cardinality induced connected subgraph $H$ of $G$ such that $H$ is color-balanced, 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 one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithm for the Steiner Tree problem on both the interval graphs and circular arc graphs, that is used as a subroutine for solving BCS problem on same graph classes. Finally, we present a FPT algorithm for the BCS problem on general graphs.