Crossing Minimization in Perturbed Drawings

  • Radoslav FulekEmail author
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)


Due to data compression or low resolution, nearby vertices and edges of a graph drawing may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map \(\varphi :G\rightarrow \mathbb {R}^2\). We wish to perturb \(\varphi \) by an arbitrarily small \(\varepsilon >0\) into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An \(\varepsilon \)-perturbation, for every \(\varepsilon >0\), is given by a piecewise linear map \(\psi _\varepsilon :G\rightarrow \mathbb {R}^2\) with \(\Vert \varphi -\psi _\varepsilon \Vert <\varepsilon \), where \(\Vert .\Vert \) is the uniform norm (i.e., \(\sup \) norm).

We present a polynomial-time solution for this optimization problem when G is a cycle and the map \(\varphi \) has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and \(\varphi \) has no spurs, and (ii) when \(\varphi \) may have spurs and G is a cycle or a union of disjoint paths.


Map approximation C-planarity Crossing number 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of Science and TechnologyKlosterneuburgAustria
  2. 2.California State University NorthridgeLos AngelesUSA
  3. 3.Tufts UniversityMedfordUSA

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