A matroidal criterion for flow polytopes to be order polytopes

Flow polytopes of directed acyclic graphs form a central class of lattice polytopes in algebraic, geometric, and enumerative combinatorics. Order polytopes are one of the best understood families of lattice polytopes; their Ehrhart theory, triangulations, volumes, and face structures are closely controlled by the combinatorics of the underlying posets. M\'esz\'aros--Morales--Striker proved that the flow polytope of an $st$-planar directed acyclic graph is unimodularly equivalent to an order polytope. In this paper, we prove a converse after contracting idle edges. More precisely, for a directed acyclic graph $G$ with a unique source and a unique sink, let $\widetilde G$ be the graph obtained from $G$ by successively contracting idle edges until none remain. We prove that $\mathcal{F}(G)$ is unimodularly equivalent to an order polytope if and only if $\widetilde G$ is $st$-planar. In addition, under a local three-good-neighbor condition, we prove that for a directed acyclic graph with a unique source, a unique sink, and no idle edges, the graph is $st$-planar if and only if it avoids an explicit list of forbidden butterfly minors.