# Graph Bases and Diagram Commutativity

Original Paper

## Abstract

Given two cycles A and B in a graph, such that $$A\cap B$$ is a non-trivial path, the connected sum $$A\hat{+} B$$ 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.

## Keywords

Cycle basis Connected sum Commutative diagram Groupoid Robust cycle basis Ear basis Geodesic cycle

## Mathematics Subject Classification

05C38 20L05 18A10

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© Springer Japan KK, part of Springer Nature 2018

## Authors and Affiliations

1. 1.Department of Mathematics, Box 2014Virginia Commonwealth UniversityRichmondUSA
2. 2.Department of Mathematics and StatisticsGeorgetown UniversityWashingtonUSA