Let $G=(V,E)$ be an undirected unweighted multi-graph and $S\subseteq V$ be a subset of vertices. A set of edges with the least cardinality whose removal disconnects $S$, that is, there is no path between at least one pair of vertices from $S$, is called a Steiner mincut for $S$ or simply an $S$-mincut. Connectivity Carcass is a compact data structure storing all $S$-mincuts in $G$ announced by Dinitz and Vainshtein in an extended abstract by Dinitz and Vainshtein in 1994. The complete proof of various results of this data structure for the simpler case when the capacity of $S$-mincut is odd appeared in the year 2000 in SICOMP. Over the last couple of decades, there have been attempts towards the proof for the case when the capacity of $S$-mincut is even, but none of them met a logical end. We present the following results. - We present the first complete, self-contained exposition of the connectivity carcass which covers both even and odd cases of the capacity of $S$-mincut. - We derive the results using an alternate and much simpler approach. In particular, we derive the results using submodularity of cuts -- a well-known property of graphs expressed using a simple inequality. - We also show how the connectivity carcass can be helpful in efficiently answering some basic queries related to $S$-mincuts using some additional insights.

翻译：令 $G=(V,E)$ 为一个无向无权多重图，$S\subseteq V$ 为顶点子集。一个基数最小的边集，其移除后使 $S$ 不连通（即 $S$ 中至少有一对顶点之间不存在路径），称为 $S$ 的斯坦纳最小割，或简称为 $S$ 最小割。连通性骨架是一种紧凑数据结构，用于存储 $G$ 中的所有 $S$ 最小割，由 Dinitz 和 Vainshtein 于 1994 年在一篇扩展摘要中提出。在 $S$ 最小割容量为奇数的较简单情形下，该数据结构各项结果的完整证明于 2000 年发表在 SICOMP 上。在过去的几十年间，人们曾尝试证明 $S$ 最小割容量为偶数的情形，但均未取得逻辑上的圆满结果。我们提出以下成果：- 我们首次给出了连通性骨架的完整且自包含的阐述，涵盖了 $S$ 最小割容量的奇数和偶数两种情形。- 我们采用另一种更简单的方法推导出结果。具体而言，我们利用割的子模性（一种通过简单不等式刻画的图的著名性质）进行推导。- 我们还展示了如何借助一些额外的洞察，利用连通性骨架高效地回答与 $S$ 最小割相关的基本查询。