Graph Bases and Diagram Commutativity View Full Text


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Article Info

DATE

2018-07

AUTHORS

Richard H. Hammack, Paul C. Kainen

ABSTRACT

Given two cycles A and B in a graph, such that A∩B is a non-trivial path, the connected sumA+^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. More... »

PAGES

523-534

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00373-018-1891-y

DOI

http://dx.doi.org/10.1007/s00373-018-1891-y

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1103160701


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