Graphs and Combinatorics

, Volume 34, Issue 4, pp 523–534 | Cite as

Graph Bases and Diagram Commutativity

  • Richard H. Hammack
  • Paul C. Kainen
Original Paper


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.


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

Mathematics Subject Classification

05C38 20L05 18A10 



We thank the referees for helpful comments.


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Copyright information

© 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

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