  We study the \emph{sensitivity oracles problem for subgraph connectivity} in
the \emph{decremental} and \emph{fully dynamic} settings. In the fully dynamic
setting, we preprocess an $n$-vertices $m$-edges undirected graph $G$ with
$n_{\rm off}$ deactivated vertices initially and the others are activated. Then
we receive a single update $D\subseteq V(G)$ of size $|D| = d \leq d_{\star}$,
representing vertices whose states will be switched. Finally, we get a sequence
of queries, each of which asks the connectivity of two given vertices $u$ and
$v$ in the activated subgraph. The decremental setting is a special case when
there is no deactivated vertex initially, and it is also known as the
\emph{vertex-failure connectivity oracles} problem.
  We present a better deterministic vertex-failure connectivity oracle with
$\widehat{O}(d_{\star}m)$ preprocessing time, $\widetilde{O}(m)$ space,
$\widetilde{O}(d^{2})$ update time and $O(d)$ query time, which improves the
update time of the previous almost-optimal oracle [Long-Saranurak, FOCS 2022]
from $\widehat{O}(d^{2})$ to $\widetilde{O}(d^{2})$.
  We also present a better deterministic fully dynamic sensitivity oracle for
subgraph connectivity with $\widehat{O}(\min\{m(n_{\rm off} +
d_{\star}),n^{\omega}\})$ preprocessing time, $\widetilde{O}(\min\{m(n_{\rm
off} + d_{\star}),n^{2}\})$ space, $\widetilde{O}(d^{2})$ update time and
$O(d)$ query time, which significantly improves the update time of the state of
the art [Hu-Kosinas-Polak, 2023] from $\widetilde{O}(d^{4})$ to
$\widetilde{O}(d^{2})$. Furthermore, our solution is even almost-optimal
assuming popular fine-grained complexity conjectures.
