J. Engelfriet and J. J. Vereijken have shown that linear graph grammars based on hyperedge replacement and regular graph expressions can be considered as interpretations of linear string grammars. In this paper we show that finite graph automata can be lifted from strings to graphs within the same framework. For the efficient recognition of graphs, we define how finite graph automata can be factorized by a modified powerset construction, and state sufficient conditions, uniques start nodes and free edge choice, under which factorized graph automata recognize graphs predictively, without backtracking, and in linear time.