  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 a strongly-connected (i.e., $k=1$) digraph $G$, we show that whether
$V$ is in (a) and/or (b) can be decided in $O(n+m)$ time and that 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).
