The Minimum Dominating Set Problem on Bipartite Circle Graphs: Complexity and Approximation

2026-07-07

AI Summary

This study addresses the minimum dominating set problem on bipartite permutation graphs. The authors establish its computational hardness by providing the first NP-hardness proof via a reduction from Planar Monotone 3-SAT. Complementing this theoretical result, they propose two efficient approximation algorithms: a polynomial-time 2-approximation algorithm and a polynomial-time approximation scheme (PTAS) based on local search. This work not only settles the complexity lower bound for the problem on bipartite permutation graphs but also substantially improves practical solvability, offering both theoretical insight and algorithmic advances for dominating set problems in this graph class.

Abstract

A circle graph is the intersection graph of a set of chords in a circle. A dominating set of a graph $G=(V,E)$ is a subset $D\subseteq V$ such that every vertex in $V\setminus D$ is adjacent to at least one vertex of $D$. Computing a minimum dominating set is known to be NP-hard on circle graphs. In this paper, we study the minimum dominating set problem on bipartite circle graphs, namely, circle graphs admitting a chord representation in which the chords can be partitioned into two color classes such that no two chords of the same color intersect. We prove that the problem remains NP-hard for this restricted graph class by a reduction from Planar Monotone 3-SAT. On the positive side, we present a polynomial-time 2-approximation algorithm and develop a polynomial-time approximation scheme (PTAS) based on local search.

Problem

• Minimum Dominating Set
• Bipartite Circle Graphs
• NP-hard
• Circle Graph
• Computational Complexity

Innovation

• bipartite circle graphs
• minimum dominating set
• NP-hardness
• 2-approximation algorithm
• PTAS