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