Graph cycles and diagram commutativity

Bases are exhibited for Kn, Kp,q, and Qd, and it is shown how each cycle of the various graphs can be built as a hierarchical ordered sum in which all of the partial sums are (simple) cycles with each cycle from either the basis or one of the hierarchically constructed cycle-sets meeting the partial sum of its predecessors in a nontrivial path. A property that holds for this “connected sum” of two cycles whenever it holds for both the parents is called constructable. It is shown that any constructable property holding for the specified basis cycles holds for every cycle in the graph, that commutativity is a constructable property of cycles in a groupoid diagram, and that “economies of scale” apply to ensuring commutativity for diagrams of the above three types. A procedure is given to extend a commutative groupoid diagram for any digraph that contains the diagram’s scheme.