For which graphs does every edge belong to exactly two chordless cycles? (Q1918870)

Summary: A graph is 2-cycled if each edge is contained in exactly two of its chordless cycles. The 2-cycled graphs arise in connection with the study of balanced signing of graphs and matrices. The concept of balance of a \(\{0, +1, -1\}\)-matrix or a signed bipartite graph has been studied by Truemper and by Conforti et al. The concept of \(\alpha\)-balance is a generalization introduced by Truemper. Truemper exhibits a family \({\mathcal F}\) of planar graphs such that a graph \(G\) can be signed to be \(\alpha\)-balanced if and only if each induced subgraph of \(G\) in \({\mathcal F}\) can. We show here that the graphs in \({\mathcal F}\) are exactly the 2-connected 2-cycled graphs.