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Crossing Minimization in Perturbed Drawings

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

Abstract

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.

Keywords

Map approximation C-planarity Crossing number 

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Copyright information

© 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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