relations on graphs

[John Stell <jgs@comp.leeds.ac.uk>]

For a set, X, relations on X are equivalent to
join-preserving functions on the powerset P(X).

If we replace X by a graph, the usual notion
of a relation on a graph is  a pair of relations
one on edges and one on nodes subject to an
obvious compatibility condition. However such
relations are not as general as the join-preserving
functions on the bi-Heyting algebra of subgraphs
(consider for example the one node, one edge graph).
If we mean relations in this more general sense
could there be a notion of converse? (anything
for which R** = R, and 1* = 1, and (RS)* = S*R*)

Is there any literature which discusses different
possible notions for relations on graphs?

John Stell