Column Planarity and Partial Simultaneous Geometric Embedding

  • William Evans
  • Vincent Kusters
  • Maria Saumell
  • Bettina Speckmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any ε > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + ε)n.

We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partial SGE (PSGE). A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • William Evans
    • 1
  • Vincent Kusters
    • 2
  • Maria Saumell
    • 3
  • Bettina Speckmann
    • 4
  1. 1.University of British ColumbiaCanada
  2. 2.Department of Computer ScienceETH ZürichSwitzerland
  3. 3.Department of Mathematics and European Centre of Excellence NTISUniversity of West BohemiaCzech Republic
  4. 4.Technical University EindhovenThe Netherlands

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