  In this note, we revisit the edge-degree constrained subgraph (EDCS)
introduced by Bernstein and Stein (ICALP'15). An EDCS is a sparse subgraph
satisfying simple edge-degree constraints that is guaranteed to include an
(almost) $\frac{2}{3}$-approximate matching of the base graph. Since its
introduction, the EDCS has been successfully applied to numerous models of
computation. Motivated by this success, we revisit EDCS and present an improved
bound for its key property in general graphs.
  Our main result is a new proof of the approximation guarantee of EDCS that
builds on the graph's Gallai-Edmonds decomposition, avoiding the probabilistic
method of the previous proofs. As a result, we get that to obtain a
$(\frac{2}{3} - \epsilon)$-approximation, a sparse EDCS with maximum degree
bounded by $O(1/\epsilon)$ is sufficient. This improves the
$O(\log(1/\epsilon)/\epsilon^2)$ bound of Assadi and Bernstein (SOSA'19) and
the $O(1/\epsilon^3)$ bound of Bernstein and Stein (SODA'16). Our guarantee
essentially matches what was previously only known for bipartite graphs,
thereby removing the gap in our understanding of EDCS in general vs. bipartite
graphs.
