二分嫁接中的奇割 II：去资本距离分量的结构与普适性
Odd Cuts in Bipartite Grafts II: Structure and Universality of Decapital Distance Components
AI总结：
本文研究二分嫁接中T-割最大打包，刻画去资本距离分量的结构，并给出两顶点下该分量对根不变的充要条件，进而证明其总数等于最小联结边数的两倍。
摘要：
本文是刻画二分嫁接中T-割最大打包的一系列论文中的第二篇，继第一篇论文（N. Kita，“二分嫁接中的紧割 I：资本距离分量”，arXiv:2202.00192v2，2022）之后。给定一个嫁接$(G, T)$、一个最小联结$F$以及一个称为根的特指顶点$r$，$(G, T)$的距离分量定义为由$F$诱导的距离所确定的$G$的子图。若一个距离分量包含根，则称为资本分量；否则称为去资本分量。在我们的第一篇论文中，我们研究了二分嫁接中资本距离分量的规范结构，该结构可用Kotzig–Lovász分解的嫁接类比来描述。在本文中，我们为去资本距离分量提供了对应的结构。我们还建立了两个顶点$r$和$r'$的一个充分必要条件，使得相对于根$r$的去资本距离分量也相对于根$r'$成为去资本距离分量。作为推论，我们得到：在二分嫁接中，对所有根的选择取遍后，去资本距离分量的总数等于该嫁接的最小联结中边数的两倍。
英文摘要：
This paper is the second in a series of papers characterizing the maximum packing of \( T \)-cuts in bipartite grafts, following the first paper (N.~Kita, ``Tight cuts in bipartite grafts~I: Capital distance components,'' {arXiv:2202.00192v2}, 2022). Given a graft $(G, T)$, a minimum join $F$, and a specified vertex $r$ called the root, the distance components of $(G, T)$ are defined as subgraphs of $G$ determined by the distances induced by $F$. A distance component is called {\em capital} if it contains the root; otherwise, it is called {\em decapital}. In our first paper, we investigated the canonical structure of capital distance components in bipartite grafts, which can be described using the graft analogue of the Kotzig--Lovász decomposition. In this paper, we provide the counterpart structure for the decapital distance components. We also establish a necessary and sufficient condition for two vertices $r$ and $r'$ under which a decapital distance component with respect to root $r$ is also a decapital distance component with respect to root $r'$. As a consequence, we obtain that the total number of decapital distance components in a bipartite graft, taken over all choices of root, is equal to twice the number of edges in a minimum join of the graft.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
