The complexity of computing the cylindrical and the $t$-circle crossing number of a graph

TL;DR

This paper proves that computing the cylindrical and t-circle crossing numbers of a graph are NP-complete problems, settling an open question about their computational complexity.

Contribution

It establishes the NP-completeness of calculating the cylindrical and t-circle crossing numbers, providing a significant complexity result in graph drawing theory.

Findings

Proves cylindrical crossing number computation is NP-complete.

Shows t-circle crossing number computation is NP-complete.

Addresses an open problem listed by Schaefer.

Abstract

A plane drawing of a graph is {\em cylindrical} if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The {\em cylindrical crossing number} of a graph \(G\) is the minimum number of crossings in a cylindrical drawing of \(G\). In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper we settle this by showing that this problem is NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number, namely the \(t\)-{\em circle crossing number}.