A new perspective on clustered planarity as a combinatorial embedding problem

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• 作者：Thomas Blä ; sius ; ^{blaesius@kit.edu" class="auth_mail" title="E-mail the corresponding author} ; Ignaz Rutter ; ¹ ; ^{rutter@kit.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Graph drawing ; Clustered planarity ; Constrained planar embedding ; Characterization ; Algorithms
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：4 January 2016
• 年：2016
• 卷：609
• 期：part_P2
• 页码：306-315
• 全文大小：467 K

文摘

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges.

Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.