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arXiv · 1510.03152

Treetopes and their Graphs

David Eppstein
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Abstract

We define treetopes, a generalization of the three-dimensional roofless polyhedra (Halin graphs) to arbitrary dimensions. Like roofless polyhedra, treetopes have a designated base facet such that every face of dimension greater than one intersects the base in more than one point. We prove an equivalent characterization of the 4-treetopes using the concept of clustered planarity from graph drawing, and we use this characterization to recognize the graphs of 4-treetopes in polynomial time. This result provides one of the first classes of 4-polytopes, other than pyramids and stacked polytopes, that can be recognized efficiently from their graphs.

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David Eppstein. 2015-10-12. Treetopes and their Graphs. https://doi.org/10.1007/s00454-020-00177-0

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