  We prove tight network topology dependent bounds on the round complexity of
computing well studied $k$-party functions such as set disjointness and element
distinctness. Unlike the usual case in the CONGEST model in distributed
computing, we fix the function and then vary the underlying network topology.
This complements the recent such results on total communication that have
received some attention. We also present some applications to distributed graph
computation problems.
  Our main contribution is a proof technique that allows us to reduce the
problem on a general graph topology to a relevant two-party communication
complexity problem. However, unlike many previous works that also used the same
high level strategy, we do not reason about a two-party communication problem
that is induced by a cut in the graph. To `stitch' back the various lower
bounds from the two party communication problems, we use the notion of timed
graph that has seen prior use in network coding. Our reductions use some tools
from Steiner tree packing and multi-commodity flow problems that have a delay
constraint.
