Column Planarity and Partial Simultaneous Geometric Embedding

Evans, William S. and Kusters, Vincent and Saumell, Maria and Speckmann, Bettina (2014) Column Planarity and Partial Simultaneous Geometric Embedding. In: Graph Drawing 22nd International Symposium, GD 2014, September 24-26, 2014 , pp. 259-271(Official URL:

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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.

Item Type: Conference Paper
Additional Information: 10.1007/978-3-662-45803-7_22
Classifications: G Algorithms and Complexity > G.490 Embeddings
G Algorithms and Complexity > G.700 Layering
M Methods > M.300 Dynamic / Incremental / Online
M Methods > M.900 Tree
P Styles > P.480 Layered
P Styles > P.540 Planar
P Styles > P.720 Straight-line
P Styles > P.999 Others
Z Theory > Z.750 Topology

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