Hi Doug and Mira:

you were both questioning:

does G contain a subgraph homeomorphic to H

versus

is H homeomorphic to G.

Below I argue that we want the first. Do you agree?

-- Karl
=========
Given strategies

S2 = {A -> B -> C -> D} with Source(S2)=A and Source(S2)=D
S1 = {B -> X -> C} with Source(S1)=B and Source(S1)=C

do we view S1 a substrategy of S2.

Yes, especially if we want to use strategies as interface strategy types.

Definition:
A strategy S1 is a substrategy of strategy S2, if
there is a subgraph S3 of S2 such that
for all class graphs G:
PathSet[G](S1) is a subset of PathSet[G](S3).      

This seems to correspond directly to subgraph homeomorphism:

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?      
===============================================
Hi Johan:

does the following theorem hold?
How would you prove it?

This is needed to decide whether a strategy is of a certain strategy
type.

-- Karl
=====================

Definition:
A strategy s1 is a substrategy of strategy s2, if 
for all class graphs G:
PathSet[G](s1) is a subset of PathSet[G](s2).

Theorem:
s1 is a substrategy of s2 iff PathSet[s1](s2) is not empty.

-- Karl