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Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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


Given a bipartite graph G = (Vb, Vr, E), the 2-Level Quasi-Planarity problem asks for the existence of a drawing of G in the plane such that the vertices in Vb and in Vr lie along two parallel lines b and r, respectively, each edge in E is drawn in the unbounded strip of the plane delimited by b and 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 Dujmović, Pór, and Wood. Furthermore, we show that the problem becomes linear-time solvable if the ordering of the vertices in Vb along 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.

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cover image Proceedings
Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 2779 - 2798
Editor: Dániel Marx, CISPA Helmholtz Center for Information Security, Germany
ISBN (Online): 978-1-61197-646-5


Published online: 7 January 2021




This research was supported in part by MIUR Project “AHeAD” under PRIN 20174LF3T8, by H2020-MSCA-RISE project 734922 – “CONNECT”, and by Roma Tre University Azione 4 Project “GeoView”.

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