  The Kahan discretization of the Lotka-Volterra system, associated with any
skew-symmetric graph $\Gamma$, leads to a family of rational maps, parametrized
by the step size. When these maps are Poisson maps with respect to the
quadratic Poisson structure of the Lotka-Volterra system, we say that the graph
$\Gamma$ has the Kahan-Poisson property. We show that if $\Gamma$ is connected,
it has the Kahan-Poisson property if and only if it is a cloning of a graph
with vertices $1,2,\dots,n$, with an arc $i\to j$ precisely when $i<j$, and
with all arcs having the same value. We also prove a similar result for
augmented graphs, which correspond with deformed Lotka-Volterra systems and
show that the obtained Lotka-Volterra systems and their Kahan discretizations
are superintegrable as well as Liouville integrable.
