If
is a spanning tree of a connected, finite, undiected
graph ,
then the fundamental cycle-set of corresponding to is the set of cycles of , each consisting of one edge of together with the unique path in (Paton 1969).

There are exactly

fundamental cycles in a graph, namely one for each edge that does not belong to the spanning tree. Here, is the circuit rank, is the edge
count,
the vertex count, and the number of connected components. A set of fundamental cycles are illustrated above for the
cuboctahedral graph, together with the graph
itself and the spanning tree used to construct the
basis.

Any cycle of a graph can be expressed as a sum modulo 2 over a fundamental cycle set of the graph (Gould 1959, Paton 1969).

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