I would like to decompose undirected graph into a minimum number of paths and cycles that are edge-disjoint.
My idea is to first take longest paths, but it is not polynomial.
Do you know any polynomial algorithm?
Decompose undirected graph to minimum paths and cycles
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You might be interested in the "chain decomposition" of a graph, as described by Jens Schmidt.
It's mentioned in this wikipedia article on Ear Decompositions. I've implemented it myself, and it's a nice simple algorithm.