AI 中文总结
研究有向无环图流多胞体与序多胞体的关系，核心方法是收缩闲置边，主要贡献为证明\(\mathcal{F}(G)\)与序多胞体单模等价当且仅当\(\widetilde G\)是\(st -\)平面图，以及在特定条件下给出图是\(st -\)平面图的充要条件。
AI 中文摘要
有向无环图的流多胞体是代数、几何和枚举组合学中格多胞体的核心类别。序多胞体是最易理解的格多胞体族之一。Mészáros--Morales--Striker证明了\(st -\)平面图有向无环图的流多胞体与序多胞体是单模等价的。本文在收缩闲置边后证明了一个逆命题。具体而言，对于有唯一源和唯一汇的有向无环图\(G\)，经收缩闲置边得到\(\widetilde G\)，我们证明\(\mathcal{F}(G)\)与序多胞体单模等价当且仅当\(\widetilde G\)是\(st -\)平面图。此外，在局部三良邻条件下，对于有唯一源、唯一汇且无闲置边的有向无环图，该图是\(st -\)平面图当且仅当它避免一个明确的禁止蝴蝶子式列表。
英文摘要
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. Mészáros--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.
Comments17 pages
