AI 中文总结
本文针对排列图、区间图和良划分弦图的最大边开放打包问题，通过定向星冲突图的最大团或划分树动态规划，分别给出\textit{O}(n^4)时间的算法。
AI 中文摘要
边开放打包是诱导匹配的一种松弛形式，其中所选边可诱导出互不相交的星型结构。我们研究排列图、区间图和良划分弦图上的最大边开放打包问题。针对前两类图，我们引入定向星冲突图，其顶点为图的有序边。我们证明该图的相容图具有自然的传递定向：排列图对应乘积序定向，区间图对应从左到右定向。每种情况下，最大边开放打包均可由相容图中的最大团（等价于最长有向路径）得到。给定对应表示后，两种算法的运行时间均为\textit{O}(n^2+m^2)≤\textit{O}(n^4)，其中\textit{n}=|V(G)|，\textit{m}=|E(G)|。针对良划分弦图，我们基于划分树给出动态规划方法，其状态利用边开放打包的端点集在每个团袋中最多含两个顶点的性质。给定划分树表示后，边开放打包数的计算时间为\textit{O}(n^4)，最优打包的重构时间也在同一时间界内。
英文摘要
Edge open packing is a relaxation of induced matching in which the selected edges may induce disjoint stars. We study the \textsc{Maximum Edge Open Packing} problem on permutation graphs, interval graphs, and well-partitioned chordal graphs. For the first two classes, we introduce an oriented star-conflict graph whose vertices are ordered edges. We prove that its compatibility graph admits a natural transitive orientation: a product-order orientation for permutation graphs and a left-to-right orientation for interval graphs. In each case, a maximum edge open packing is obtained from a maximum clique, equivalently a longest directed path, in the compatibility graph. Given the corresponding representation, both algorithms run in \(O(n^2+m^2)\leq O(n^4)\) time, where \(n=|V(G)|\) and \(m=|E(G)|\). For well-partitioned chordal graphs, we give a dynamic program over a partition tree. Its states use the fact that the endpoint set of an edge open packing meets each clique bag in at most two vertices. Given a partition-tree representation, the edge open packing number is computed in \(O(n^4)\) time, and an optimal packing can be reconstructed within the same time bound.
