In this paper, we first study what we call Superset-Subset-Disjoint (SSD) set system. Based on properties of SSD set system, we derive the following (I) to (IV): (I) For a nonnegative integer $k$ and a graph $G=(V,E)$ with $|V|\ge2$, let $X_1,X_2,\dots,X_q\subsetneq V$ denote all maximal proper subsets of $V$ that induce $k$-edge-connected subgraphs. Then at least one of (a) and (b) holds: (a) $\{X_1,X_2,\dots,X_q\}$ is a partition of $V$; and (b) $V\setminus X_1, V\setminus X_2,\dots,V\setminus X_q$ are pairwise disjoint. (II) For $k=1$ and a strongly-connected digraph $G$, whether $V$ is in (a) and/or (b) can be decided in $O(n+m)$ time and we can generate all such $X_1,X_2,\dots,X_q$ in $O(n+m+|X_1|+|X_2|+\dots+|X_q|)$ time, where $n=|V|$ and $m=|E|$. (III) For a digraph $G$, we can enumerate in linear delay all vertex subsets of $V$ that induce strongly-connected subgraphs. (IV) A digraph is Hamiltonian if there is a spanning subgraph that is strongly-connected and in the case (a).

翻译：本文首先研究了我们称之为超集-子集-不相交（SSD）的集合系统。基于SSD集合系统的性质，我们推导出以下（I）至（IV）项结果：（I）对于一个非负整数$k$和一个满足$|V|\ge2$的图$G=(V,E)$，令$X_1,X_2,\dots,X_q\subsetneq V$表示所有能导出$k$-边连通子图的$V$的极大真子集。则以下（a）和（b）至少有一项成立：（a）$\{X_1,X_2,\dots,X_q\}$构成$V$的一个划分；（b）$V\setminus X_1, V\setminus X_2,\dots,V\setminus X_q$两两不相交。（II）对于$k=1$和一个强连通有向图$G$，可在$O(n+m)$时间内判定$V$是否满足（a）和/或（b），并能在$O(n+m+|X_1|+|X_2|+\dots+|X_q|)$时间内生成所有这样的$X_1,X_2,\dots,X_q$，其中$n=|V|$，$m=|E|$。（III）对于有向图$G$，我们能够以线性延迟枚举所有能导出强连通子图的顶点子集$V$。（IV）若存在一个生成子图既是强连通的又满足情况（a），则该有向图是哈密顿图。