Graph Rewriting and Relabeling with PBPO+ (Extended Version)

We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO+, exerts more control over the embedding of the pattern in the host graph, which is important for a large class of graph rewrite systems. We show that PBPO+ is well-suited for rewriting labeled graphs and certain classes of attributed graphs. For this purpose, we employ a lattice structure on the label set and use graph morphisms that respect the partial order on the labels. We argue that our approach is both simpler and more general than related approaches in the literature. Finally, we demonstrate that PBPO+ allows for an elegant modeling of linear term rewriting systems, such as string rewriting systems. This modeling preserves termination as a global system property without having to restrict the shape of the graphs.