Treetopes and their Graphs
arXiv:1510.03152 · doi:10.1007/s00454-020-00177-0
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.
16 pages, 8 figures. To appear at 27th ACM-SIAM Symp. on Discrete Algorithms (SODA 2016)