Another formulation of the fixed point property is to say that
for every order-preserving map there is a p that has distance
0 from its image.
As this article is about algorithmic approaches
(toward which the author is strongly biased), we did not touch upon this.
However
M. Roddy's proof of the product conjecture in [104] and
the work in [82]
use this approach, showing the strength of this point of view.
It would be interesting to see how much further this
approach can be taken.
For example with respect to Xia's algorithm it is easy to see
that Xia's algorithm removes all edges
with d(a,c)<d(b,d)
from the
(fixed point free) endomorphism graph.
It also removes all edges 
for which there
is an 
that is comparable to
all elements of H from the fixed point free endomorphism graph.
What other removals can be guaranteed via similar arguments?
As a consequence, how much of a performance gain over the
depth-first search can be guaranteed?