A kuratowski-type theorem for planarity of partially embedded graphs

Authors:
Vít Jelínek;Jan Kratochvíl;Ignaz Rutter
Affiliations:
Faculty of Mathematics, University of Vienna, Vienna, Austria;Charles University, Prague, Czech Rep;Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Karlsruhe, Germany
Venue:
Proceedings of the twenty-seventh annual symposium on Computational geometry
Year:
2011

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Abstract

A partially embedded graph (or PEG) is a triple (G,H,EH), where G is a graph, H is a subgraph of G, and EH is a planar embedding of H. We say that a PEG (G,H,EH) is planar if the graph G has a planar embedding that extends the embedding EH. 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 a minimal obstruction.