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 graph m...

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Bibliographic Details
Published inComputational geometry : theory and applications Vol. 46; no. 4; pp. 466 - 492
Main Authors Jelínek, Vít, Kratochvíl, Jan, Rutter, Ignaz
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.05.2013
Subjects
Algorithms
Computational geometry
Containment
Graphs
Kuratowski theorem
Mathematical analysis
Obstructions
Partially embedded graphs
Planar graphs
Recognition
Theorems
Kuratowski theorem
Partially embedded graphs
Planar graphs
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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.
Bibliography:ObjectType-Article-2
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ISSN:0925-7721
DOI:10.1016/j.comgeo.2012.07.005