Graph rewriting and relabeling with PBPO+: A unifying theory for quasitoposes

We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO⁺, allows more control over the embedding of the pattern in the host graph, which is important for a large class of rewrite systems. We argue that PBPO⁺ can be considered a unifying theory in the general setting of quasitoposes, by demonstrating that PBPO⁺ can define a strict superset of the rewrite relations definable by PBPO, AGREE and DPO. Additionally, we show that PBPO⁺ is well suited for rewriting labeled graphs and some classes of attributed graphs, by introducing a lattice structure on the label set and requiring graph morphisms to be order-preserving.