Upward Planarity Testing of Biconnected Outerplanar DAGs Solves Partition

We show an $O(n)$-time reduction from the problem of testing whether a multiset of positive integers can be partitioned into two multisets so that the sum of the integers in each multiset is equal to $n\mathrm{/}2$ to the problem of testing whether an $n$-vertex biconnected outerplanar DAG admits an upward planar drawing. This constitutes the first barrier to the existence of efficient algorithms for testing the upward planarity of DAGs with no large triconnected minor. We also show a result in the opposite direction. Suppose that partitioning a multiset of positive integers into two multisets so that the sum of the integers in each multiset is $n\mathrm{/}2$ can be solved in $f(n)$ time. Let $G$ be an $n$-vertex biconnected outerplanar DAG and $e$ be an edge incident to the outer face of an outerplanar drawing of $G$. Then it can be tested in $O(f(n))$ time whether $G$ admits an upward planar drawing with $e$ on the outer face.