  Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of $G$ and
demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on $H$ within the network with
capacities defined on $G$. We prove a previous conjecture, which states that
when the supply graph $G$ is series-parallel, the pair $(G,H)$ is
cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of $G$ and delete edges
of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with
$p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of
degree 2, and an edge connecting the two vertices of degree $p$. We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.
