
TL;DR
This paper introduces treetopes, a new class of high-dimensional polytopes, and provides a polynomial-time method to recognize their graphs, expanding understanding of 4-polytopes beyond pyramids and stacked forms.
Contribution
It defines treetopes and characterizes 4-treetopes using clustered planarity, enabling efficient recognition of their graphs, a novel advancement in polytope graph theory.
Findings
Defined treetopes as a generalization of roofless polyhedra.
Provided polynomial-time recognition algorithm for 4-treetope graphs.
Expanded the class of 4-polytopes recognizable from their graphs.
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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