Connectivity of cubical polytopes

Abstract

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d ≥ 3, the graph of a cubical d-polytope with minimum degree δ is min{δ, 2d − 2}-connected. Second, we show, for any d ≥ 4, that every minimum separator of cardinality at most 2d − 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself.to be convex. Figure 2 depicts this operation. A connected sum of two copies of a cyclic d-polytope with d ≥ 4 and n ≥ d + 1 vertices ([15, Thm. 0.7]), which is a polytope whose facets are all simplices, results in a d-polytope of minimum degree n − 1 that is d-connected but not (d + 1)-connected.On our way to prove the connectivity theorem we prove results of independent interest, for instance, the following (Corollary 16 in Section 4).Corollary. Let P be a cubical d-polytope and let F be a proper face of P . Then the subgraphRemark 3. The examples of Fig. 1 also establish that the previous corollary is best possible in the sense that the removal of the vertices of a proper face F of a cubical d-polytope does not always leave a (d − 1)-connected subgraph of the graph of the polytope.

The linkedness of cubical polytopes: beyond the cube

“…The aim of this section is to present a couple of results related to the connectivity of strongly connected complexes in cubical polytopes. The first results are from [1].…”

“…First, using the (k -1)-linkedness of L 1 (Proposition 7) find disjoint paths L i := s i -t i in L 1 for i ∈ [2, k]. It may happen that t F 1 1 is in one of the paths L i for i ∈ [2, k]. Second, consider neighbours s 1 and t 1 in A 1 of s 1 and t F 1 1 , respectively.…”

“…Second, consider neighbours s 1 and t 1 in A 1 of s 1 and t F 1 1 , respectively. If t F 1 1 doesn't belong to any path L i then find a path L 1 := s 1 -t 1 that contains the edge t 1 t F 1 1 and a subpath L 1 in A 1 between s 1 and t 1 ; that is,…”

“…, L k }. If t F 1 1 belongs to one of the paths L i with i ∈ [2, k], say L j , then disregard this path L j and consider in A 1 a neighbour s j of s j and a neighbour t j of t j . From Lemma 25, it follows that the vertices s 1 , t 1 , s j and t j can be taken pairwise distinct.…”

“…Our methodology relies on results on the connectivity of strongly connected subcomplexes of cubical polytopes, whose proof ideas were first developed in [1], and a number of new insights into the structure of d-cube exposed in [2]. One obstacle that forces some tedious analysis is the fact that the 3-cube is not 2-linked.…”