3 problems

Let $P$ be a simple $d$-polytope with $m$ vertices, and let $G_P$ be its graph. Kalai's separator conjecture. There exists a subset $V'$ of vertices of $P$ such that … and removing…

Let $P$ be a polytope, let $F$ be a facet of $P$, and let $C$ be a cutset. Facet-connectedness conjecture. The cutset $C$ cannot contain vertices $u$ and $v$ lying in a common face…

Let $P$ be a $d$-polytope with graph $G_P$, and let a cutset be a set of vertices selected for removal by a hyperplane. Properties 1 and 2 of Proposition are connectedness of the c…