Recently graph data models have become increasingly popular in many scientific fields. Efficient query processing over such data is critical. Existing works often rely on index structures that store pre-computed transitive relations to achieve efficient graph matching. In this paper, we present a family of stack-based algorithms to handle path and twig pattern queries for directed acyclic graphs (DAGs) in particular. With the worst-case space cost linearly bounded by the number of edges in the graph, our algorithms achieve a quadratic runtime complexity in the average size of the query variable bindings. This is optimal among the navigation-based graph matching algorithms.