Deterministic Fully Dynamic Data Structures for Vertex Cover and   Matching
  We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m}\,)$ time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a $(2+\eps)$ approximation in $O(\log n/\eps^2)$ amortized time per update.   For maximum matching, we show how to maintain a $(3+\eps)$ approximation in $O(\min(\sqrt{n}/\epsilon, m^{1/3}/\eps^2))$ {\em amortized} time per update, and a $(4+\eps)$ approximation in $O(m^{1/3}/\eps^2)$ {\em worst-case} time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld from STOC' 2010. 
Sayan Bhattacharya and Monika Henzinger and Giuseppe F. Italiano
cs.DS