Polygon-Universal Graphs

Authors Tim Ophelders, Ignaz Rutter , Bettina Speckmann , Kevin Verbeek

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Author Details

Tim Ophelders
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Ignaz Rutter
  • Department of Computer Science and Mathematics, Universität Passau, Germany
Bettina Speckmann
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Kevin Verbeek
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands


Ignaz Rutter would like to thank Michael Hoffmann and Vincent Kusters for discussions on conjectures related to this paper.

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Tim Ophelders, Ignaz Rutter, Bettina Speckmann, and Kevin Verbeek. Polygon-Universal Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study a fundamental question from graph drawing: given a pair (G,C) of a graph G and a cycle C in G together with a simple polygon P, is there a straight-line drawing of G inside P which maps C to P? We say that such a drawing of (G,C) respects P. We fully characterize those instances (G,C) which are polygon-universal, that is, they have a drawing that respects P for any simple (not necessarily convex) polygon P. Specifically, we identify two necessary conditions for an instance to be polygon-universal. Both conditions are based purely on graph and cycle distances and are easy to check. We show that these two conditions are also sufficient. Furthermore, if an instance (G,C) is planar, that is, if there exists a planar drawing of G with C on the outer face, we show that the same conditions guarantee for every simple polygon P the existence of a planar drawing of (G,C) that respects P. If (G,C) is polygon-universal, then our proofs directly imply a linear-time algorithm to construct a drawing that respects a given polygon P.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Graph drawing
  • partial drawing extension
  • simple polygon


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