Forbidding Kuratowski Graphs as Immersions

Abstract : The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting from graphs that are planar sub-cubic or of branch-width at most 10.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00904533
Contributor : Dimitrios M. Thilikos <>
Submitted on : Thursday, November 14, 2013 - 4:14:06 PM
Last modification on : Friday, October 5, 2018 - 9:14:01 PM

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Archontia C. Giannopoulou, Marcin Kaminski, Dimitrios M. Thilikos. Forbidding Kuratowski Graphs as Immersions. Journal of Graph Theory, Wiley, 2015, 78 (1), pp.43-60. ⟨10.1002/jgt.21790⟩. ⟨lirmm-00904533⟩

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