Hanani-Tutte for approximating maps of graphs Conference Paper


Author(s): Fulek, Radoslav; Kynčl, Jan
Title: Hanani-Tutte for approximating maps of graphs
Title Series: Leibniz International Proceedings in Information, LIPIcs
Affiliation IST Austria
Abstract: We resolve in the affirmative conjectures of A. Skopenkov and Repovš (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.
Keywords: computational geometry; Graph theory; Computation theory; Planarity; Clustered planarity; Hanani-Tutte theorem; Graph embeddings; Graph embedding; Map approximation; Weak embedding; Piecewise linear techniques
Conference Title: SoCG: Symposium on Computational Geometry
Volume: 99
Conference Dates: June 11 - 14, 2018
Conference Location: Budapest, Hungary
ISBN: 18688969 (ISSN); 9783959770668 (ISBN)
Publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik  
Date Published: 2018-01-01
Start Page: 391
End Page: 3915
DOI: 10.4230/LIPIcs.SoCG.2018.39
Open access: yes (repository)
IST Austria Authors
  1. Radoslav Fulek
    14 Fulek
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