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# Minimizing Crossings in Constrained Two-Sided Circular Graph Layouts

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## Cite As

Fabian Klute and Martin Nöllenburg. Minimizing Crossings in Constrained Two-Sided Circular Graph Layouts. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.53

## Abstract

Circular layouts are a popular graph drawing style, where vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is NP-hard. One way to allow for fewer crossings in practice are two-sided layouts that draw some edges as curves in the exterior of the circle. In fact, one- and two-sided circular layouts are equivalent to one-page and two-page book drawings, i.e., graph layouts with all vertices placed on a line (the spine) and edges drawn in one or two distinct half-planes (the pages) bounded by the spine. In this paper we study the problem of minimizing the crossings for a fixed cyclic vertex order by computing an optimal k-plane set of exteriorly drawn edges for k >= 1, extending the previously studied case k=0. We show that this relates to finding bounded-degree maximum-weight induced subgraphs of circle graphs, which is a graph-theoretic problem of independent interest. We show NP-hardness for arbitrary k, present an efficient algorithm for k=1, and generalize it to an explicit XP-time algorithm for any fixed k. For the practically interesting case k=1 we implemented our algorithm and present experimental results that confirm the applicability of our algorithm.
##### Keywords
• Graph Drawing
• Circular Layouts
• Crossing Minimization
• Circle Graphs
• Bounded-Degree Maximum-Weight Induced Subgraphs

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