Аннотация:Toric topology assigns to each n-dimensional combinatorial simple convex polytope P with m facets an (m+n)-dimensional moment-angle manifold Z_P with an action of the compact torus Tm such that Z_P /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 each simple n-polytope Q such that the graded rings H^∗(Z_P,Z) and H^∗(Z_Q,Z) are isomorphic 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 L^3. Pogorelov polytopes include fullerenes—simple polytopes with onlypentagonal and hexagonal faces. It is known that the properties to be flag and to be Pogorelov are B-rigid. We focus on almost Pogorelov polytopes, which are strongly cyclically 4-edge-connected polytopes. They correspond toright-angled polytopes of finite volume in 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 to be an ideal almost Pogorelov polytope are B-rigid. As a corollary, we obtain that the 3-dimensional associahedron As^3 and permutohedron Pe^3 are B-rigid. We generalize methods known for Pogorelov polytopes. We obtain results on B-rigidityof subsets in H^∗(Z_P,Z) and prove an analog of the so-called separable circuit condition (SCC).