We report progress on dynamic complexity of well-known graph problems such as reachability and matching. In this model, edges are dynamically added and deleted and we measure the complexity of each update/query. We propose polynomial-size data structures for such updates for several related problems. The updates are in very low level complexity classes such as quantifier-free first order formulas, AC~0, TC~0. In particular, we show the following problems are in the indicated classes: (a) maximum matching in non-uniform DynTC~0; (b) digraph reachability in non-uniform DynAC~0; (c) embedded planar digraph reachability in DynFO(= uniform DynAC~0). Notably, the part (c) of our results yields the first non-trivial class of graphs where reachability can be maintained by first-order updates; it is a long-standing open question whether the same holds for general graphs. For (a) we show that the technique in can in fact be generalized using and to maintain the determinant of a matrix in DynTC~0. For (b) we extend this technique with the help of two more ingredients namely isolation and truncated approximation using rational polynomials. In fact, our proof yields DynAC~0 [p] bound for any prime p ＞ 1. For (c) we exploit the duality between cuts and cycles in planar graphs to maintain the number of crossings between carefully chosen primal and dual paths, using several new structural lemmas.
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