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Polyline Drawings with Topological Constraints

Authors Emilio Di Giacomo , Peter Eades, Giuseppe Liotta , Henk Meijer, Fabrizio Montecchiani

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Author Details

Emilio Di Giacomo
  • Università degli Studi di Perugia, Perugia, Italy
Peter Eades
  • University of Sydney, Sydney, Australia
Giuseppe Liotta
  • Università degli Studi di Perugia, Perugia, Italy
Henk Meijer
  • University College Roosevelt, Middelburg, The Netherlands
Fabrizio Montecchiani
  • Università degli Studi di Perugia, Perugia, Italy

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Emilio Di Giacomo, Peter Eades, Giuseppe Liotta, Henk Meijer, and Fabrizio Montecchiani. Polyline Drawings with Topological Constraints. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 39:1-39:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


Let G be a simple topological graph and let Gamma be a polyline drawing of G. We say that Gamma partially preserves the topology of G if it has the same external boundary, the same rotation system, and the same set of crossings as G. Drawing Gamma fully preserves the topology of G if the planarization of G and the planarization of Gamma have the same planar embedding. We show that if the set of crossing-free edges of G forms a connected spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Omega(sqrt{n}). Concerning drawings that fully preserve the topology, we show that if G has skewness k, it admits one such drawing with curve complexity at most 2k; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Theory of computation → Design and analysis of algorithms
  • Topological graphs
  • graph drawing
  • curve complexity
  • skewness-k graphs
  • k-planar graphs


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