How to efficiently determine whether a given ladder is valid?
This is not a complete answer.
I give a simpler statement of the problem
and some remarks.
We start with a graph where vertices are labeled with $[n]$.
We have an operation that adds a directed edge from $v$ to $u$ to the graph,
and if $label(v)&lt;label(u)$ switches their labels.
Given a directed multi-graph $G$ with $n$ vertices and $e$ edges,
check if it can be obtained using using the operation above.
It is easy to see that the problem is in $\mathsf{NP}$:
a certificate is a (polynomial size) sequence of operations resulting in the $G$.
Observation
It seems that we can assume without loss of generality that
all edges to the last vertex are added at the end of the sequence and
all edges from it are added at the start of the sequence.
This can be generalized to other vertices.
Assume that we have remove all vertices with labels larger than $label(v)$.
All edges to $v$ are added at the end of the sequence and
all edges from $v$ are added at the start of the sequence.
I think it should be possible to combined this observation with Havel-Hakimi to give a polynomial time algorithm.