found that:
:A graph G is factor-critical if and only if G has an odd ear decomposition.More generally, a result of makes it possible to find in any graph G the ear decomposition with the fewest even ears.

Series-parallel graphs

A tree ear decomposition is a proper ear decomposition in which the first ear is a single edge and for each subsequent ear , there is a single ear , , such that both endpoints of lie on . A nested ear decomposition is a tree ear decomposition such that, within each ear , the set of pairs of endpoints

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