Graph Bases and Diagram Commutativity

Given two cycles A and B in a graph, such that A ∩ B is a non-trivial path, the connected sum A + ^ B is the cycle whose edges are the symmetric difference of E ( A ) and E ( B ). A special kind of cycle basis for a graph, a connected sum basis , is defined. Such a basis has the property that a hierarchical method, building successive cycles through connected sum, eventually reaches all the cycles of the graph. It is proved that every graph has a connected sum basis. A property is said to be cooperative if it holds for the connected sum of two cycles when it holds for the summands. Cooperative properties that hold for the cycles of a connected sum basis will hold for all cycles in the graph. As an application, commutativity of a groupoid diagram follows from commutativity of a connected sum basis for the underlying graph of the diagram. An example is given of a noncommutative diagram with a (non-connected sum) basis of cycles which do commute.