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.

Dedicated to the memory of Alexander Rubinov, an outstanding mathematician, a great person, teacher and friend, on the occasion of his 80 th birthday

The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is $k$-linked if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is $k$-linked if its graph is $k$-linked. We establish that the $d$-dimensional cube is $\lfloor (d+1)/2 \rfloor$-linked, for every $d\ne 3$; this is the maximum possible linkedness of a $d$-polytope. This result implies that, for every $d\geqslant 1$, a cubical $d$-polytope is  $\lfloor d/2\rfloor$-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009).  Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph $G$ is strongly $k$-linked if it has at least $2k+1$ vertices and, for  every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We show that cubical 4-polytopes are strongly $2$-linked and that, for each $d\geqslant 1$,  $d$-dimensional cubes  are strongly $\lfloor d/2\rfloor$-linked.

In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings.

The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analysed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining 'extremal' statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.