  The $k$ red domination problem for a bipartite graph $G=(X,Y,E)$ is to find a
subset $D \subseteq X$ of cardinality at most $k$ that dominates vertices of
$Y$. The decision version of this problem is NP-complete for general bipartite
graphs but solvable in polynomial time for chordal bipartite graphs. We
strengthen that result by showing that it is NP-complete for perfect
elimination bipartite graphs. We present a tight upper bound on the number of
such sets in bipartite graphs, and show that we can calculate that number in
linear time for convex bipartite graphs. We present a linear space linear delay
enumeration algorithm that needs only linear preprocessing time.
