Graph Rewriting and Relabeling with PBPO$$^{+}$$

We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO $$^{+}$$ , exerts more control over embedding of pattern in host graph, which is important a large class rewrite systems. In addition, we show that well-suited labeled graphs and certain classes attributed graphs. For this purpose, employ lattice structure on label set use order-preserving morphisms. argue our simpler general than related relabeling approaches literature.