  We consider a function computation problem in a three node wireless network.
Nodes A and B observe two correlated sources $X$ and $Y$ respectively, and want
to compute a function $f(X,Y)$. To achieve this, nodes A and B send messages to
a relay node C at rates $R_A$ and $R_B$ respectively. The relay C then
broadcasts a message to A and B at rate $R_C$. We allow block coding, and study
the achievable region of rate triples under both zero-error and
$\epsilon$-error. As a preparation, we first consider a broadcast network from
the relay to A and B. A and B have side information $X$ and $Y$ respectively.
The relay node C observes both $X$ and $Y$ and broadcasts an encoded message to
A and B. We want to obtain the optimal broadcast rate such that A and B can
recover the function $f(X,Y)$ from the received message and their individual
side information $X$ and $Y$ respectively. For this problem, we show
equivalence between $\epsilon$-error and zero-error computations-- this gives a
rate characterization for zero-error computation. As a corollary, this also
gives a rate characterization for the relay network under zero-error for a
class of functions called {\em component-wise one-to-one functions} when the
support set of $p_{XY}$ is full. For the relay network, the zero-error rate
region for arbitrary functions is characterized in terms of graph coloring of
some suitably defined probabilistic graphs. We then give a single-letter inner
bound to this rate region. Further, we extend the graph theoretic ideas to
address the $\epsilon$-error problem and obtain a single-letter inner bound.
