  Matrix partition problems generalize a number of natural graph partition
problems, and have been studied for several standard graph classes. We prove
that each matrix partition problem has only finitely many minimal obstructions
for split graphs. Previously such a result was only known for the class of
cographs. (In particular, there are matrix partition problems which have
infinitely many minimal chordal obstructions.) We provide (close) upper and
lower bounds on the maximum size of a minimal split obstruction. This shows for
the first time that some matrices have exponential-sized minimal obstructions
of any kind (not necessarily split graphs). We also discuss matrix partitions
for bipartite and co-bipartite graphs.
