Confluence up to garbage in graph transformation

• Generalising confluence to up garbage. Critical pair analysis for Back-tracking free recognition of graph languages. The transformation graphs and graph-like structures is ubiquitous in computer science. When a system described by graph-transformation rules, it often desirable that the rules are both terminating confluent so rule applications an arbitrary order produce unique resulting graphs. However, there application scenarios where not globally but on subclass interest. In other words, non-resolvable conflicts can only occur considered as “garbage”. this paper, we introduce notion garbage generalise Plump's critical lemma double-pushout transformation, providing sufficient condition non-garbage analysis. We apply our results two case studies about efficient language recognition: present backtracking-free reduction systems which recognise class flow diagrams labelled series-parallel graphs, respectively. Both non-confluent also give subcommutativity which, together with closedness, implies even non-terminating systems.