Treetopes and their Graphs

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.