  This work addresses Byzantine vector consensus (BVC), wherein the input at
each process is a d-dimensional vector of reals, and each process is expected
to decide on a decision vector that is in the convex hull of the input vectors
at the fault-free processes [3, 8]. The input vector at each process may also
be viewed as a point in the d-dimensional Euclidean space R^d, where d > 0 is a
finite integer. Recent work [3, 8] has addressed Byzantine vector consensus in
systems that can be modeled by a complete graph. This paper considers Byzantine
vector consensus in incomplete graphs. In particular, we address a particular
class of iterative algorithms in incomplete graphs, and prove a necessary
condition, and a sufficient condition, for the graphs to be able to solve the
vector consensus problem iteratively. We present an iterative Byzantine vector
consensus algorithm, and prove it correct under the sufficient condition. The
necessary condition presented in this paper for vector consensus does not match
with the sufficient condition for d > 1; thus, a weaker condition may
potentially suffice for Byzantine vector consensus.
