On removable cycles through every edge
Mader and Jackson independently proved that every 2-connected simple graph G with minimum degree at least four has a removable cycle, that is, a cycle C such that G\E(C) is 2-connected. This paper considers the problem of determining when every edge of a 2-connected graph G, simple or not, can be guaranted to lie in some removable cycle. The main result establishes that if every deletion of two edges from G remains 2-connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge-disjoint removable cycles such that each contains one of the distinguished edges.
Publication Source (Journal or Book title)
Journal of Graph Theory
Lemos, M., & Oxley, J. (2003). On removable cycles through every edge. Journal of Graph Theory, 42 (2), 155-164. https://doi.org/10.1002/jgt.10082