A key problem in processing graph-based meaning representations is graph parsing, i.e. computing all possible derivations of a given graph according to a (competence) grammar. We demonstrate, for the first time, that exact graph parsing can be efficient for large graphs and with large Hyperedge Replacement Grammars (HRGs). The advance is achieved by exploiting locality as terminal edge-adjacency in HRG rules. In particular, we highlight the importance of 1) a terminal edge-first parsing strategy, 2) a categorization of a subclass of HRG, i.e. what we call Weakly Regular Graph Grammar, and 3) distributing argument-structures to both lexical and phrasal rules.