Graph bases and diagram commutativity

A diagram is a digraph where vertices and arcs are objects and morphisms in a category. When all morphisms are invertible, the underlying digraph of the diagram is a graph. A cycle is said to commute if the composition of its morphisms equals the identity. We exhibit a diagram of homeomorphisms which commutes on all the cycles of a basis for the underlying graph but is not commutative. The diagram is an orientation of the complete bipartite graph K_{3,3}. We show that every graph has a type of basis (called a CS-basis) with the property that if a diagram of isomorphisms commutes on all the cycles of a CS-basis for the underlying graph of a diagram, then it commutes for all cycles of the diagram. Applications to data models will be discussed.