$B$-rigidity of the property to be an almost Pogorelov polytope

Toric topology assigns to each $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold ${\mathcal{Z}}_P$ with an action of a compact torus $T^m$ such that ${\mathcal{Z}}_P\mathrm{/}{T}^m$ is a convex polytope of combinatorial type $P$. We study the notion of $B$-rigidity. A property of a polytope $P$ is called $B$-rigid, if any isomorphism of graded rings $H^($\mathcal{Z}$_P,\mathbb Z)= H^($\mathcal{Z}$_Q,\mathbb Z)$forasimple$n$-polytope$Q$ implies that it also has this property. We study families of $3$-dimensional polytopes defined by their cyclic $k$-edge-connectivity. These families include flag polytopes and Pogorelov polytopes, that is polytopes realizable as bounded right-angled polytopes in Lobachevsky space ${\mathbb{L}}^3$. Pogorelov polytopes include fullerenes – simple polytopes with only pentagonal and hexagonal faces. It is known that the properties to be flag and Pogorelov polytope are $B$-rigid. We focus on almost Pogorelov polytopes, which are strongly cyclically $4$-edge-connected polytopes. They correspond to right-angled polytopes of finite volume in ${\mathbb{L}}^3$. There is a subfamily of ideal almost Pogorelov polytopes corresponding to ideal right-angled polytopes. We prove that the properties to be an almost Pogorelov polytope and an ideal almost Pogorelov polytope are $B$-rigid. As a corollary we obtain that $3$-dimensional associahedron $A{s}^3$ and permutohedron $P{e}^3$ are $B$-rigid. We generalize methods known for Pogorelov polytopes. We obtain results on $B$-rigidity of subsets in $H^{\ast }({\mathcal{Z}}_P,\mathbb{Z})$ and prove an analog of the so-called separable circuit condition (SCC). As an example we consider the ring $H^{\ast }({\mathcal{Z}}_{A{s}^3},\mathbb{Z})$.