On simultaneous planar graph embeddings

Authors:
Peter Brass;Eowyn Cenek;Cristian A. Duncan;Alon Efrat;Cesim Erten;Dan P. Ismailescu;Stephen G. Kobourov;Anna Lubiw;Joseph S. B. Mitchell
Affiliations:
Department of Computer Science, City College of New York, USA;Department of Computer Science, University of Waterloo, USA;Department of Computer Science, University of Miami, USA;Department of Computer Science, University of Arizona, USA;Department of Computer Science, University of Arizona, USA;Department of Mathematics, Hofstra University, USA;Department of Computer Science, University of Arizona, USA;Department of Computer Science, University of Waterloo, USA;Department of Applied Mathematics and Statistics, Stony Brook University, USA
Venue:
Computational Geometry: Theory and Applications
Year:
2007

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Abstract

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an nxn grid, and two caterpillar graphs on a 3nx3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)xO(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n^2)xO(n^2) grid.