The concept of total dual integrality dates back to the works of
Edmonds, Giles and Pulleyblank in the late 70's, and is strongly
connected to min-max relations in combinatorial optimization.
In this work we show a characterization of series-parallel graphs in
terms of box-total dual integrality of the k-edge-connected spanning
subgraph polyhedron.

The system ${A}x\ge b$ is totally dual integral (TDI) if, for each
integer vector c for which $\min \{c x:Ax\ge b\}$ is finite, there
exists an integer optimal solution of $\max \{yb:yA = c,\, y \ge0\}$
such that:
$$\min \{c x:Ax\ge b\} = \max \{yb:yA=c, \,y \ge0\}.$$
It is known that every integer polyhedron can be described by a TDI
system $Ax\ge b$ with $A$ and $b$ integer. The integrality of the TDI
system is desirable because, then we have a min-max relation between
combinatorial objects.

We are interested in the stronger property of box-TDIness. A system $
A x\ge  b$ is called box-TDI if the system $Ax\ge b, \ell\le x \le u$
is TDI for all rational vectors $\ell$ and $u$.
A polyhedron that can be described by box-TDI system is called a
box-TDI polyhedron.
This definition is motivated by the fact that any TDI system
describing a box-TDI polyhedron is box-TDI.
The past few years, this property has received a renewed interest and
several new box-TDI systems were discovered.

We prove that, for $k\ge2$, the k-edge-connected spanning subgraph
polyhedron is a box-TDI polyhedron if and only if the graph is
series-parallel. Moreover, in this case, we provide a box-TDI system
with integer coefficients describing this polyhedron.