  We study the multi-agent formation control problem in a directed graph. We
show that a desired relative configuration scheme is reasonable in a directed
graph if and only if for any cycle in this directed graph, the product of
relative configurations of the forward arcs, and inverses of relative
configurations of the backward arcs, is equal to $1$. We then show that for a
reasonable desired relative configuration scheme in a directed and connected
graph, the dual quaternion Laplacian is similar to the unweighted Laplacian of
the directed graph, zero is an eigenvalue of these two Laplacian matrices, and
a formation vector is a desired formation satisfying this reasonable desired
relative configuration scheme if and only if it is a unit dual quaternion
vector and an eigenvector, corresponding to the zero eigenvalue, of the dual
quaternion Laplacian matrix. A numerical method is given to determine {whether}
a given desired relative configuration scheme is reasonable or not. A control
law is presented when the desired relative configuration scheme is reasonable.
Our discussion is extended to general directed gain graphs.
