A Kuratowski-Type Theorem for Planarity of Partially Embedded Graphs
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2012
- DOI:
- 10.48550/arXiv.1204.2915
- arXiv:
- arXiv:1204.2915
- Bibcode:
- 2012arXiv1204.2915J
- Keywords:
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- Computer Science - Discrete Mathematics;
- G.2.2
- E-Print:
- 45 pages, 18 figures