Column Planarity and Partially-Simultaneous Geometric Embedding

0483679 - ÚI 2018 RIV US eng J - Journal Article
Barba, L. - Evans, W. - Hoffmann, M. - Kusters, V. - Saumell, Maria - Speckmann, B.
Column Planarity and Partially-Simultaneous Geometric Embedding.
Journal of Graph Algorithms and Applications. Roč. 21, č. 6 (2017), s. 983-1002. ISSN 1526-1719
Grant - others:GA MŠk(CZ) LO1506; GA MŠk(CZ) EE2.3.30.0038
Institutional support: RVO:67985807
Keywords : column planarity * unlabeled level planarity * simultaneous geometric embedding
OECD category: Pure mathematics

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 the maximum size of column planar subsets of trees: every tree on n vertices contains a column planar set of size at least 14n/17 and for any epsilon > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + epsilon)n. In addition, we show that every outerplanar graph has a column planar set of size at least n/2. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (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, which are PSGEs in which at least k vertices are mapped to the same point for both graphs. In particular, we show that every two trees on n vertices admit an 11n/17-PSGE and every two outerplanar graphs admit an n/4-PSGE.
Permanent Link: http://hdl.handle.net/11104/0278897