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Definition: A path is simple if it does not contain any edge twice:
A path is elementary if it does not contain any node twice:
p is simple : <=> forall i in Nn, j in Nn: <pi, pi+1> = <pj, pj+1> => i = j where n = length(p).
A path is a cycle or circuit if it terminates in its initial node:
p is elementary : <=> (forall i in Nn, j in Nn: pi = pj => i = j) where n = 1+length(p).
p is cycle : <=> exists x: p is path from x to x.