relations on graphs

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