Theory of Computing Systems

, Volume 61, Issue 4, pp 1054–1083 | Cite as

Dynamic Planar Embeddings of Dynamic Graphs

  • Jacob Holm
  • Eva Rotenberg


We present an algorithm to support the dynamic embedding in the plane of a dynamic graph. An edge can be inserted across a face between two vertices on the face boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices u,v, linkable(u,v) decides whether u and v are linkable in the current embedding, and if so, returns a list of suggestions for the placement of (u,v) in the embedding. For non-linkable vertices u,v, we define a new query, one-flip- linkable(u,v) providing a suggestion for a flip that will make them linkable if one exists. We support all updates and queries in \(\mathcal {O}(\log ^{2} n)\) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph. Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant interaction between top trees over the two trees via their common Euler tour.


Data structures Planar graphs Graph algorithms Graph embeddings Dynamic data structures Graph theory 



We would like to thank Christian Wulff-Nilsen and Mikkel Thorup for many helpful and interesting discussions and ideas.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark

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