  In their breakthrough ICALP'15 paper, Bernstein and Stein presented an
algorithm for maintaining a $(3/2+\epsilon)$-approximate maximum matching in
fully dynamic {\em bipartite} graphs with a {\em worst-case} update time of
$O_\epsilon(m^{1/4})$; we use the $O_\epsilon$ notation to suppress the
$\epsilon$-dependence. Their main technical contribution was in presenting a
new type of bounded-degree subgraph, which they named an {\em edge degree
constrained subgraph (EDCS)}, which contains a large matching -- of size that
is smaller than the maximum matching size of the entire graph by at most a
factor of $3/2+\epsilon$. They demonstrate that the EDCS can be maintained with
a worst-case update time of $O_\epsilon(m^{1/4})$, and their main result
follows as a direct corollary. In their followup SODA'16 paper, Bernstein and
Stein generalized their result for general graphs, achieving the same update
time of $O_\epsilon(m^{1/4})$, albeit with an amortized rather than worst-case
bound. To date, the best {\em deterministic} worst-case update time bound for
{\em any} better-than-2 approximate matching is $O(\sqrt{m})$ [Neiman and
Solomon, STOC'13], [Gupta and Peng, FOCS'13]; allowing randomization (against
an oblivious adversary) one can achieve a much better (still polynomial) update
time for approximation slightly below 2 [Behnezhad, Lacki and Mirrokni,
SODA'20].
  In this work we\footnote{\em quasi nanos, gigantium humeris insidentes}
simplify the approach of Bernstein and Stein for bipartite graphs, which allows
us to generalize it for general graphs while maintaining the same bound of
$O_\epsilon(m^{1/4})$ on the {\em worst-case} update time. Moreover, our
approach is {\em density-sensitive}: If the {\em arboricity} of the dynamic
graph is bounded by $\alpha$ at all times, then the worst-case update time of
the algorithm is $O_\epsilon(\sqrt{\alpha})$.
