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arXiv · 1207.5329

Forbidding Kuratowski Graphs as Immersions

Archontia C. Giannopoulou·Marcin Kaminski·Dimitrios M. Thilikos
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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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Archontia C. Giannopoulou, Marcin Kaminski, Dimitrios M. Thilikos. 2012-07-23. Forbidding Kuratowski Graphs as Immersions. https://arxiv.org/abs/1207.5329

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