A Kuratowski-Type Theorem for Planarity of Partially Embedded Graphs

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 gr...

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Bibliographic Details
Published in:arXiv.org 2012-04
Main Authors: Jelínek, Vít, Kratochvíl, Jan, Rutter, Ignaz
Format: Article
Language:English
Subjects:
Algorithms
Containment
Embedding
Graph theory
Polynomials
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Summary: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.
ISSN:2331-8422