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.

翻译：我们扩展了具有强匹配功能的拉回-推出（PBPO）图重写方法。所提出的方法称为PBPO+，能更精细地控制模式在宿主图中的嵌入过程，这对一大类重写系统至关重要。通过证明PBPO+可定义PBPO、AGREE和DPO所定义重写关系的严格超集，我们认为PBPO+在拟拓扑的广义框架下可视为一种统一理论。此外，通过引入标签集上的格结构并要求图态射保持序关系，我们证明了PBPO+特别适用于标号图及某些带属性图的重写。