2-Level Quasi-Planarity or How Caterpillars Climb (SPQR-)Trees

TL;DR

This paper proves that the 2-Level Quasi-Planarity problem is NP-complete, but becomes efficiently solvable with a fixed vertex order, advancing understanding of recognizing quasi-planar graphs.

Contribution

It establishes the NP-completeness of recognizing 2-Level Quasi-Planar graphs and provides a linear-time algorithm for fixed vertex order cases, using SPQR-trees and caterpillar structures.

Findings

NP-completeness of 2-Level Quasi-Planarity

Linear-time solvability with fixed vertex order

First results on recognizing quasi-planar graphs

Abstract

Given a bipartite graph $G=(V_b,V_r,E)$, the $2$-Level Quasi-Planarity problem asks for the existence of a drawing of $G$ in the plane such that the vertices in $V_b$ and in $V_r$ lie along two parallel lines $\ell_b$ and $\ell_r$, respectively, each edge in $E$ is drawn in the unbounded strip of the plane delimited by $\ell_b$ and $\ell_r$, and no three edges in $E$ pairwise cross. We prove that the $2$-Level Quasi-Planarity problem is NP-complete. This answers an open question of Dujmovi\'c, P\'{o}r, and Wood. Furthermore, we show that the problem becomes linear-time solvable if the ordering of the vertices in $V_b$ along $\ell_b$ is prescribed. Our contributions provide the first results on the computational complexity of recognizing quasi-planar graphs, which is a long-standing open question. Our linear-time algorithm exploits several ingredients, including a combinatorial characterization of the positive instances of the problem in terms of the existence of a planar embedding with a caterpillar-like structure, and an SPQR-tree-based algorithm for testing the existence of such a planar embedding. Our algorithm builds upon a classification of the types of embeddings with respect to the structure of the portion of the caterpillar they contain and performs a computation of the realizable embedding types based on a succinct description of their features by means of constant-size gadgets.