L O ] 5 A pr 2 01 7 Reachability is in DynFO

Patnaik and Immerman introduced the dynamic complexity class DynFO of database queries that can be maintained by first-order dynamic programs with the help of auxiliary relations under insertions and deletions of edges [34]. This article confirms their conjecture that the Reachability query is in DynFO. As a byproduct it is shown that the rank of a matrix with small values can be maintained in DynFO(+,×). It is further shown that the (size of the) maximum matching of a graph can be maintained in non-uniform DynFO, another extension of DynFO, with non-uniform initialisation of the auxiliary relations.