There are several "morphism" definitions between graphs.
Which one is closest to "substrategy"?

From Garey and Johnson:

subgraph homeomorphism problem (OPEN 2)
instance: graph G = (V,E)
question: does G contain a subgraph homeomorphic to H, i.e.,
a subgraph G'=(V',E') that can be converted to a graph isomorphic to H by
repeatedly removing any vertex of degree 2
and adding the edge joining its two neighbors?         

graph contractability (GT51)
instance: graphs G=(V1,E1) and H=(V2,E2)
question: can a graph isomorphic to H be obtained from G by a 
sequence of edge contractions, i.e. a sequence in which each step
replaces two adjacent vertices u,v by a single vertex w adjacent to exactly
those vertices that were previously adjacent to at least one of u and v?

Is NP-complete but can be solved in poly. time if H is a triangle.

There is also homomorphism and digraph D-morphism but
subgraph homeomorphism problem is the one we want.