Realizations of branched self-coverings of the 2-sphere

For a degree-d   branched self-covering of the 2-sphere, a notable combinatorial invariant is an integer partition of 2d−22d−2, consisting of the multiplicities of the critical points. A finer invariant is the so-called Hurwitz passport. The realization problem of Hurwitz passports remains largely open till today. In this article, we introduce two different types of finer invariants: a bipartite map and an incidence matrix. We then settle completely their realization problem by showing that a bipartite map, or a matrix, is realized by a branched covering if and only if it satisfies a certain balanced condition. A variant of the bipartite map approach was initiated by W. Thurston. Our results shed some new light to the Hurwitz passport problem.