非嵌套完美匹配的最小阻断集
Minimum blockers and extremal graphs for nonnested matchings
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中文总结 AI 辅助
本文分类了有序图上非嵌套完美匹配的最小阻断集，给出三个显式族共$2^k+k-2$个成员，并提供了$O(k^2)$算法及对非完美情形的推广。
中文摘要 AI 辅助
有序图中的完美匹配称为非嵌套的，如果没有任何一条边严格位于另一条边的内部。我们对$2k$个有序顶点上所有非嵌套完美匹配的最小边集进行了分类。对于$k\ge2$，这些阻断集有$k$条边，属于三个显式族，总共有$2^k+k-2$个成员。证明使用了辅助区间割的最小生成树；其等号情形也对交叉匹配的串联的最小阻断集进行了分类。该论证给出一个确定性算法，从至多$k$条被删除的边出发，在$O(k^2)$次字操作和$O(k)$个辅助字内返回一个最小阻断集描述或一个避免的非嵌套完美匹配。我们还给出了从一个阻断集族到$123$-避免置换矩阵的最小阻断集的单射。在完美情形之外，一个显式构造给出了具有$(k-1)n+1$条边且对每个$k\ge5$和$n\ge2k+1$没有非嵌套$k$-匹配的图。
英文摘要
We classify all smallest sets of edge deletions that destroy every nonnested perfect matching in a complete ordered graph. The vertices have a fixed linear order. A perfect matching selects edges that use each vertex exactly once; it is nonnested when no selected edge has both endpoints strictly between those of another. On $2k$ vertices, exactly $k$ deletions are necessary, and we describe all $2^k+k-2$ minimum deletion sets for every $k\ge2$. Allowing a matching to leave vertices unused changes the extremal problem. Barát, Freschi and Tóth proposed that an $n$-vertex ordered graph avoiding a nonnested $k$-edge matching can have at most $(k-1)n$ edges. We construct counterexamples for every $k\ge5$ and $n\ge2k+1$. For $n\ge3k$, our constructions exceed the proposed value by a number of edges proportional to $k^2$, matching the order of the known upper bound on this excess. The classification follows from cuts between consecutive intervals; the larger constructions coordinate deletions at the two ends of the vertex order. The exact extremal value remains open in general.
发表机构
- Centro de Informática, Universidade Federal de Pernambuco（巴西国立伯南布哥大学信息学中心）
机构由 AI 辅助整理，请以论文原文为准。
