  We study error exponents for the problem of low-rate communication over a
directed graph, where each edge in the graph represents a noisy communication
channel, and there is a single source and destination. We derive maxflow-based
achievability and converse bounds on the error exponent that match when there
are two messages and all channels satisfy a symmetry condition called pairwise
reversibility. More generally, we show that the upper and lower bounds match to
within a factor of 4. We also show that with three messages there are cases
where the maxflow-based error exponent is strictly suboptimal, thus showing
that our tightness result cannot be extended beyond two messages without
further assumptions.
