In 1933 Whitney described the relationship between pairs of graphs with the same set of cycles (here we view cycles as sets of edges). A natural question is to try to understand the relationship between pairs of graphs with the same set of even cycles (a cycle is even if it contains an even number of edges). We show that there is a nice answer to this question under some connectivity assumptions and we explain the relevance of this question to matroid theory.

 

This is joint work with: Cheolwon Heo, Zouhaier Ferchiou, Irene Pivotto.