  Computing the Strongly-Connected Components (SCCs) in a graph $G=(V,E)$ is
known to take only $O(m + n)$ time using an algorithm by Tarjan from
1972[SICOMP 72] where $m = |E|$, $n=|V|$. For fully-dynamic graphs, conditional
lower bounds provide evidence that the update time cannot be improved by
polynomial factors over recomputing the SCCs from scratch after every update.
Nevertheless, substantial progress has been made to find algorithms with fast
update time for \emph{decremental} graphs, i.e. graphs that undergo edge
deletions.
  In this paper, we present the first algorithm for general decremental graphs
that maintains the SCCs in total update time $\tilde{O}(m)$, thus only a
polylogarithmic factor from the optimal running time. Previously such a result
was only known for the special case of planar graphs [Italiano et al, STOC
2017]. Our result should be compared to the formerly best algorithm for general
graphs achieving $\tilde{O}(m\sqrt{n})$ total update time by Chechik et.al.
[FOCS 16] which improved upon a breakthrough result leading to $O(mn^{0.9 +
o(1)})$ total update time by Henzinger, Krinninger and Nanongkai [STOC 14,
ICALP 15]; these results in turn improved upon the longstanding bound of
$O(mn)$ by Roditty and Zwick [STOC 04].
  All of the above results also apply to the decremental Single-Source
Reachability (SSR) problem, which can be reduced to decrementally maintaining
SCCs. A bound of $O(mn)$ total update time for decremental SSR was established
already in 1981 by Even and Shiloach [JACM 1981].
  Using a well known reduction, we can maintain the reachability of pairs $S
\times V$, $S \subseteq V$ in fully-dynamic graphs with update time
$\tilde{O}(\frac{|S|m}{t})$ and query time $O(t)$ for all $t \in [1,|S|]$; this
generalizes an earlier All-Pairs Reachability where $S = V$ [{\L}\k{a}cki, TALG
2013].
