Graph Realization for Bipartite Graphs

Determine whether a given non-decreasing sequence d = (d1, ..., dn) of natural numbers can be realized as the degree sequence of a labeled simple bipartite graph on the vertex set {v1, ..., vn}, that is, decide the bipartite variant of the Graph Realization problem asking for the existence of a bipartite realizing graph for the input sequence.

Background

The classical Graph Realization problem asks whether a non-decreasing sequence of natural numbers is the degree sequence of some labeled simple graph. This problem is solvable in polynomial time via the Erdős–Gallai characterization and constructive algorithms.

A natural variant restricts the realizing graph to a specific class, such as bipartite graphs. While degree sequence pairs for bipartitions in bipartite graphs have been studied, the decision problem of whether a single degree sequence is realizable by some bipartite graph—without a prescribed partition—has remained unresolved for decades.