We determine the structure of a class of graphs that do not contain the complete graph on five vertices as a “signed minor.” The result says that each graph in this class can be decomposed into elementary building blocks in which maximum packings by odd circuits can be found by flow or matching techniques. This allows us to actually find a largest collection of pairwise edge disjoint odd circuits in polynomial time (for general graphs this is NP-hard). Furthermore it provides an algorithm to test membership of our class of graphs.

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SIAM Journal on Discrete Mathematics
Probability, Networks and Algorithms

Conforti, M., & Gerards, B. (2007). Packing odd circuits. SIAM Journal on Discrete Mathematics, 21, 273–302.