Reconstructing 2-complexes by a nonnegative integer-valued vector

Over the last decade, hypergraphs have seen a lot of use in electrotechnical problems, network design, multiindex transportation problems, computer modeling of complex dynamical systems, representations of complex industrial control systems, and many other applied problems. For some of these problems (e.g., multiindex transportation problems), a better representation is given by 2-complexes, a special class of hypergraphs in which each edge is incident to three vertices. It is known that each hypergraph corresponds to the vector of degrees of its vertices, but the converse does not hold. In this work, we consider the problem of reconstructing (implementing, realizing) a 2-complex from an arbitrary vector.