not go into details of this topic, but brieﬂy sketch how our framework might be applied in this context. Let us consider 
vertex-structured graphs, where a subgraph can be reduced into a single vertex. Let G = (cid:3)V , E(cid:4) be the original graph and 
G1 = (cid:3)V 1, E1(cid:4), . . . , Gn = (cid:3)V n, En(cid:4) be the disjoint subgraphs to reduce. Let  v1, . . . , vn be the new vertices corresponding to 
i=1 V i) ∪{v1, . . . , vn}. It is then natural to deﬁne 
these subgraphs, i.e. the abstract graph will have the vertex set V
a transformation function  f such that  f (v) = vi if  v ∈ V i for some subgraph  Gi and  f (v) = v for all vertices not in any 
of the subgraphs. Given the requirements stated by the authors for when a subgraph may be reduced, it follows that this 
abstraction is PW(cid:14). Since  f
is obviously M↑, it also follows that it is PS↑. This only applies to ordinary paths in the graphs,
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