  We introduce a new dynamic data structure for maintaining the strongly
connected components (SCCs) of a directed graph (digraph) under edge deletions,
so as to answer a rich repertoire of connectivity queries. Our main technical
contribution is a decremental data structure that supports sensitivity queries
of the form "are $ u $ and $ v $ strongly connected in the graph $ G \setminus
w $?", for any triple of vertices $ u, v, w $, while $ G $ undergoes deletions
of edges. Our data structure processes a sequence of edge deletions in a
digraph with $n$ vertices in $O(m n \log{n})$ total time and $O(n^2 \log{n})$
space, where $m$ is the number of edges before any deletion, and answers the
above queries in constant time. We can leverage our data structure to obtain
decremental data structures for many more types of queries within the same time
and space complexity. For instance for edge-related queries, such as testing
whether two query vertices $u$ and $v$ are strongly connected in $G \setminus
e$, for some query edge $e$.
  As another important application of our decremental data structure, we
provide the first nontrivial algorithm for maintaining the dominator tree of a
flow graph under edge deletions. We present an algorithm that processes a
sequence of edge deletions in a flow graph in $O(m n \log{n})$ total time and
$O(n^2 \log{n})$ space. For reducible flow graphs we provide an $O(mn)$-time
and $O(m + n)$-space algorithm. We give a conditional lower bound that provides
evidence that these running times may be tight up to subpolynomial factors.
