relations on graphs

If one allows multiple edges with the same source and target then they certainly form a topos, namely that of presheaves over the category with 2 objects and 2 parallel nontrivial arrows. The \neg\neg-separated objects in this topos are precisely those graphs where there is at most one edge from one node to another one. The latter category is not a topos but a quasitopos. The non-full monos in this category are typical examples of epic monos which are not isos. All this can be found in Lawvere's "Qualitative distinctions between toposes of graphs". Thomas Streicher