Motivated by the exact weight perfect matching problem and recent parameterized algorithms for finding an $\ell$-th smallest perfect matching, we study structural properties of edge-weight symmetries in graphs. Recent work by El Maalouly et al. (ESA 2025) showed that excluding all perfect matchings whose weight is at most the $(\ell - 1)$-th smallest possible value in the graph requires fixing at most $2(\ell-1)$ edges in non-bipartite graphs and at most $\ell-1$ edges in bipartite graphs. A natural open question is whether fixing a single edge is always sufficient to shift the extreme (minimum or maximum) weight of a perfect matching when the global minimum and maximum weights differ. To address this, we define and analyze a hierarchy of progressively weaker edge-weight properties: node-induced weights, even walk and cycle symmetries, perfect matching equality, and the edge min-max property. We derive a basic hierarchy among these conditions and show that they become equivalent in bipartite graphs. For general graphs, we provide tight structural characterizations, based on block and tight cut decompositions, under which even cycle symmetry and perfect matching equality force node-induced weights. Finally, we resolve the motivating open question in the negative by constructing a matching-covered non-bipartite graph that satisfies the edge min-max property (every edge is contained in a minimum-weight perfect matching and a maximum-weight one) but violates perfect matching equality (all perfect matchings have the same weight). This counterexample shows that a single edge is not always sufficient to eliminate all minimum-weight or maximum-weight perfect matchings, thereby proving the tightness of the $2(\ell-1)$ bound for $\ell=2$. We also discuss extensions of this framework to $b$-factors and arborescences.

翻译：受精确权重完美匹配问题以及最近针对寻找第$\ell$小完美匹配的参数化算法的启发，我们研究了图中边权重对称性的结构性质。El Maalouly等人(ESA 2025)的最新工作表明，在图论中，排除所有权重不超过第$(\ell-1)$小可能值的完美匹配，在非二分图中最多需要固定$2(\ell-1)$条边，在二分图中最多需要固定$\ell-1$条边。一个自然的开放问题是：当全局最小和最大权重不同时，固定单条边是否总是足以改变完美匹配的极端（最小或最大）权重。为此，我们定义并分析了一个逐渐减弱的边权重性质的层次结构：节点诱导权重、偶游走和循环对称性、完美匹配相等性以及边最小最大性质。我们推导了这些条件之间的基本层次关系，并证明它们在二分图中等价。对于一般图，我们基于块分解和紧割分解提供了严格的结构刻画，在这些分解下，偶循环对称性和完美匹配相等性迫使节点诱导权重成立。最后，我们通过构造一个满足边最小最大性质（每条边都包含在最小权重完美匹配和最大权重完美匹配中）但违反完美匹配相等性（所有完美匹配具有相同权重）的匹配覆盖非二分图，以否定的方式解决了该激励性开放问题。这一反例表明，单条边并不总是足以消除所有最小权重或最大权重的完美匹配，从而证明了对于$\ell=2$时$2(\ell-1)$界的最优性。我们还讨论了该框架在$b$-因子和树形图上的扩展。