  Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a
generalization of Lefshcetz fibrations in order to describe near-symplectic
4-manifolds. We first study monodromy representations of higher sides of
genus-1 simplified broken Lefschetz fibrations. We then completely classify
diffeomorphism types of such fibrations with connected fibers and with less
than six Lefschetz singularities. In these studies, we obtain several families
of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain
all such fibrations, and determine the diffeomorphism types of the total spaces
of these fibrations. Our results are generalizations of Kas' classification
theorem of genus-1 Lefschetz fibrations, which states that the total space of a
non-trivial genus-1 Lefschetz fibration over $S^2$ is diffeomorphic to an
elliptic surface E(n), for some $n\geq 1$.
