2017/02/28 by Doolittle, Joseph, Nevo, Eran, Pineda-Villavicencio, Guillermo +2 · 2 citations
#52B05 (Primary) 52B12 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.08739
Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d-2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d>3 there are two d-polytopes with d-1 nonsimple vertices, isomorphic (d-3)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.