发表机构
University of Washington; Williams College（华盛顿大学; 威廉姆斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图上的稳定性条件与退化集，证明退化集对应图拟阵的单元素扩张，并从 Lawrence 多胞体细分构造稳定性条件，建立与可定向扩张的联系。
AI 中文摘要
设 $X$ 为节点曲线，$G$ 为与 $X$ 对偶的图。图 $G$ 上的稳定性条件是对 $G$ 的双连通顶点子集赋予整数，并满足某些期望性质。稳定性条件产生退化集，即 $G$ 的双连通子集的特定集合，但退化集也可以在不参考稳定性条件的情况下描述。我们证明图 $G$ 的退化集对应于图拟阵 $M(G)$ 的单元素扩张。图拟阵 $M(G)$ 的某些单元素扩张可以被定向为有向图拟阵 $\mathcal{M}(G)$ 的单元素扩张。有向拟阵的单元素扩张与对偶有向拟阵的 Lawrence 多胞体的细分一一对应。我们从余图拟阵 $\mathcal{M}^\ast(G)$ 的 Lawrence 多胞体的细分构造 $G$ 上的稳定性条件。最后，我们证明图 $G$ 的任意退化集 $\mathcal{D}$，若在退化集偏序集中以 $G$ 的所有双连通子集之集为下界，则对应于 $M(G)$ 的一个可定向扩张。
英文摘要
Let $X$ be a nodal curve, and $G$ be the graph dual to $X$. A stability condition on a graph $G$ is an assignment of integers to the biconnected subsets of vertices of $G$ satisfying some desired properties. Stability conditions yield degeneracy sets, which are certain collections of biconnected subsets of $G$, but degeneracy sets can also be described without a reference stability condition. We show that the degeneracy sets of a graph $G$ correspond to the single-element extensions of the graphic matroid $M(G)$. Some single-element extensions of the graphic matroid $M(G)$ can be oriented to be single-element extensions of the oriented graphic matroid $\mathcal{M}(G)$. Single-element extensions of oriented matroids are in bijection with subdivisions of the Lawrence polytope of the dual oriented matroid. We construct stability conditions on $G$ from subdivisions of the Lawrence polytope of the cographic matroid $\mathcal{M}^\ast(G)$. Finally, we show that any degeneracy set $\mathcal{D}$ of a graph $G$ that has the set of all biconnected subsets of $G$ as a lower bound in the poset of degeneracy sets corresponds to an orientable extension of $M(G)$.
