  Motivated by problems modeling the spread of infections in networks, in this
paper we explore which bipartite graphs are most resilient to widespread
infections under various parameter settings. Namely, we study bipartite
networks with a requirement of a minimum degree $d$ on one side under an
independent infection, independent transmission model. We completely
characterize the optimal graphs in the case $d=1$, which already produces
non-trivial behavior, and we give extremal results for the more general cases.
We show that in the case $d=2$, surprisingly, the optimally resilient set of
graphs includes a graph that is not one of the two "extremes" found in the case
$d=1$.
  Then, we briefly examine the case where we force a connectivity requirement
instead of a one-sided degree requirement and again, we find that the set of
the most resilient graphs contains more than the two "extremes." We also show
that determining the subgraph of an arbitrary bipartite graph most resilient to
infection is NP-hard for any one-sided minimal degree $d \ge 1$.
