## Abstract

We study the [Formula presented] (shortly, [Formula presented]) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. We are given a [Formula presented] graph G=(V,E) and an integer k. Each vertex in V is colored with either “[Formula presented]” or “[Formula presented]”. The BCS problem seeks an induced connected subgraph H of size at least k in G such that H is [Formula presented], 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 [Formula presented] in the same classes of graphs, which may be of independent interest.

Original language | English |
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Pages (from-to) | 69-80 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 929 |

DOIs | |

State | Published - 11 Sep 2022 |

Externally published | Yes |

## Keywords

- Balanced connected subgraph
- Circular-arc graphs
- Color-balanced
- Interval graphs
- NP-hard
- Outer-string graphs
- Permutation graphs
- Unit-disk graphs

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science