Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian

Authors Michael Hoffmann , Boris Klemz

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Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Zürich, Switzerland
Boris Klemz
  • Institute of Computer Science, Freie Universität Berlin, Berlin, Germany


This research was initiated during the Geometric Graphs Week 2016 at UPC, Barcelona, and reinvigorated at the Lorentz Center Workshop on Fixed-Parameter Computational Geometry 2018 in Leiden. We thank the organizers and participants for the stimulating atmosphere and, in particular, Oswin Aichholzer, Bahareh Banyassady, Philipp Kindermann, Irina Kostitsyna, and Fabian Lipp for initial discussions about the problem.

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Michael Hoffmann and Boris Klemz. Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We show that every triconnected planar graph of maximum degree five is subhamiltonian planar. A graph is subhamiltonian planar if it is a subgraph of a Hamiltonian planar graph or, equivalently, if it admits a 2-page book embedding. In fact, our result is stronger because we only require vertices of a separating triangle to have degree at most five, all other vertices may have arbitrary degree. This degree bound is tight: We describe a family of triconnected planar graphs that are not subhamiltonian planar and where every vertex of a separating triangle has degree at most six. Our results improve earlier work by Heath and by Bauernöppel and, independently, Bekos, Gronemann, and Raftopoulou, who showed that planar graphs of maximum degree three and four, respectively, are subhamiltonian planar. The proof is constructive and yields a quadratic time algorithm to obtain a subhamiltonian plane cycle for a given graph. As one of our main tools, which might be of independent interest, we devise an algorithm that, in a given 3-connected plane graph satisfying the above degree bounds, collapses each maximal separating triangle into a single edge such that the resulting graph is biconnected, contains no separating triangle, and no separation pair whose vertices are adjacent.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graphs and surfaces
  • Theory of computation → Computational geometry
  • Human-centered computing → Graph drawings
  • Graph drawing
  • book embedding
  • Hamiltonian graph
  • planar graph
  • bounded degree graph
  • graph augmentation
  • computational geometry
  • SPQR decomposition


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