Vertex-facet incidences of unbounded polyhedra (Q2725173)

Focusing on the question of how much information about the combinatorial structure of a polyhedron is encoded in its vertex-facet incidences, the authors show that it determines the polyhedron's dimension if the polyhedron has a bounded facet. Besides, if the polyhedron is not a cone, the vertex-facet incidences encode whether the dimension of the polyhedron is 3 or greater than 3; it also determines whether or not the polyhedron has a bounded facet. Moreover, the information whether the polyhedron is bounded is encoded too in the vertex-facet incidences. However, several examples are given that show that the combinatorial structure of a \(d\)-polyhedron is not determined by its vertex-facet incidences, but if the polyhedron is simple, i.e., if every vertex of the polyhedron is contained in exactly \(d\) facets, then the vertex-facet incidences encode its combinatorial structure. On the other hand, a \(d\)-polyhedron is called simplicial if every of its facets has precisely \(d\) vertices. The authors show that a simple and simplicial \(d\)-polyhedron is bounded and that simple and simplicial polyhedra can be characterized as those whose vertex-facet incidences matrix is a \((n,d)\) circular matrix \(M\), i.e, an \(n\times n\) 0/1 matrix whose coefficients \(m_{i,j}\) are defined as \(m_{i,j} = 1\) if \( j \in \{ i, i+1, \dots , i+d-1 \}_{\bmod n}\) and \(m_{i,j} = 0\) otherwise.