As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, at the expense of possibly producing an erroneous output on some ``yes'' instances but with very low error rate.
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"An Efficient Algorithm to Test Potential Bipartiteness of Graphical Degree Sequences,"
Theory and Applications of Graphs: Vol. 8:
1, Article 3.
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol8/iss1/3