Bounded Embeddings of Graphs in the Plane

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

A drawing in the plane (\(\mathbb {R}^2\)) of a graph \(G=(V,E)\) equipped with a function \(\gamma : V \rightarrow \mathbb {N}\) is x-bounded if (i) \(x(u) <x(v)\) whenever \(\gamma (u)<\gamma (v)\) and (ii) \(\gamma (u)\le \gamma (w)\le \gamma (v)\), where \(uv\in E\) and \(\gamma (u)\le \gamma (v)\), whenever \(x(w)\in x(uv)\), where x(.) denotes the projection to the x-axis. We prove a characterization of isotopy classes of embeddings of connected graphs equipped with \(\gamma \) in the plane containing an x-bounded embedding. Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest. This partially answers a question raised recently by Angelini et al. and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.

Keywords

Graph planarity testing Weakly simple embedding c-planarity PQ-tree Algebraic crossing number 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.IST AustriaKlosterneuburgAustria

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