Column Planarity and Partial Simultaneous Geometric Embedding
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.
KeywordsPlanar Graph Outerplanar Graph Marked Vertex Unbounded Face Unmarked Vertex
- 2.Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization (2013)Google Scholar
- 8.Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Wismath, S.: Planar and quasi planar simultaneous geometric embedding. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 52–63. Springer, Heidelberg (2014)Google Scholar