  We address the problem of reaching consensus in the presence of Byzantine
faults. In particular, we are interested in investigating the impact of
messages relay on the network connectivity for a correct iterative approximate
Byzantine consensus algorithm to exist. The network is modeled by a simple
directed graph. We assume a node can send messages to another node that is up
to $l$ hops away via forwarding by the intermediate nodes on the routes, where
$l\in \mathbb{N}$ is a natural number. We characterize the necessary and
sufficient topological conditions on the network structure. The tight
conditions we found are consistent with the tight conditions identified for
$l=1$, where only local communication is allowed, and are strictly weaker for
$l>1$. Let $l^*$ denote the length of a longest path in the given network. For
$l\ge l^*$ and undirected graphs, our conditions hold if and only if $n\ge
3f+1$ and the node-connectivity of the given graph is at least $2f+1$ , where
$n$ is the total number of nodes and $f$ is the maximal number of Byzantine
nodes; and for $l\ge l^*$ and directed graphs, our conditions is equivalent to
the tight condition found for exact Byzantine consensus.
  Our sufficiency is shown by constructing a correct algorithm, wherein the
trim function is constructed based on investigating a newly introduced minimal
messages cover property. The trim function proposed also works over
multi-graphs.
