On Hardness of the Joint Crossing Number

TL;DR

This paper proves that finding the minimal joint crossing number for two graphs embedded on a surface is NP-hard, even for surfaces of genus 6, highlighting the computational difficulty of this problem.

Contribution

It establishes NP-hardness for all variants of the joint crossing number problem on surfaces of genus 6, using a reduction from a known NP-hard problem.

Findings

All variants are NP-hard on genus 6 surfaces.

Reduction from anchored crossing number problem demonstrates complexity.

Highlights computational challenges in surface graph embeddings.

Abstract

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar.