Is every cycle basis fundamental?
David Hartvigsen, Eitan Zemel · Journal of Graph Theory · 1989
Abstract A collection of (simple) cycles in a graph is called fundamental if they form a basis for the cycle space and if they can be ordered such that Cj(C1 U … U Cj‐1) ≠ Ø for all j. We characterize by excluded minors those graphs for which every cycle basis is fundamental. We also give a constructive characterization that leads to a (polynomial) algorithm for recognizing these graphs. In addition, this algorithm can be used to determine if a graph has a cycle basis that covers every edge two or more times. An equivalent dual characterization for the cutset space is also given.