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From planar graphs to higher dimension

Abstract : In this thesis we look for generalizations of some properties of planar graphs to higher dimensions by replacing graphs by simplicial complexes.In particular we study the Dushnik-Miller dimension which measures how a partial order is far from being a linear order.When applied to simplicial complexes, this dimension seems to capture some geometric properties.In this idea, we disprove a conjecture asserting that any simplicial complex of Dushnik-Miller dimension at most d+1 can be represented as a TD-Delaunay complex in RR d, which is a variant of the well known Delaunay graphs in the plane.We show that any supremum section, particular simplicial complexes related to the Dushnik-Miller dimension, is collapsible, which means that it is possible to reach the single point by removing in a certain order the faces of the complex.We introduce the notion of stair packings and we prove that the Dushnik-Miller dimension is connected to contact complexes of such packings.We also prove new results on planar graphs.The two following theorems about representations of planar graphs are proved: any planar graph is an llcorner-intersection graph and any triangle-free planar graph is an {llcorner, | , -}-contact graph.We introduce and study a new notion on planar graphs called Möbius stanchion systems which is related to questions about unicellular embeddings of planar graphs.
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https://tel.archives-ouvertes.fr/tel-02931761
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Submitted on : Monday, September 7, 2020 - 11:04:12 AM
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  • HAL Id : tel-02931761, version 1

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Lucas Isenmann. From planar graphs to higher dimension. Discrete Mathematics [cs.DM]. Université Montpellier, 2019. English. ⟨NNT : 2019MONTS142⟩. ⟨tel-02931761⟩

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