A Kuratowski-type theorem for planarity of partially embedded graphs

Authors:
VíT JelíNek;Jan KratochvíL;Ignaz Rutter
Affiliations:
Computer Science Institute, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic;Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic;Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Box 6980, D-76128 Karlsruhe, Germany
Venue:
Computational Geometry: Theory and Applications
Year:
2013

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Abstract

A partially embedded graph (or Peg) is a triple (G,H,H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a Peg(G,H,H) is planar if the graph G has a planar embedding that extends the embedding H. We introduce a containment relation of Pegs analogous to graph minor containment, and characterize the minimal non-planar Pegs with respect to this relation. We show that all the minimal non-planar Pegs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar Pegs. Furthermore, by extending an existing planarity test for Pegs, we obtain a polynomial-time algorithm which, for a given Peg, either produces a planar embedding or identifies an obstruction.