In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify the 4-manifolds admitting genus one simplified broken Lefschetz fibrations up to blow-ups.
R. İnanç BAYKUR. Seiichi KAMADA. "Classification of broken Lefschetz fibrations with small fiber genera." J. Math. Soc. Japan 67 (3) 877 - 901, July, 2015. https://doi.org/10.2969/jmsj/06730877