Abstract: We present a graph algorithm to find fundamental cycles aligned with the principal curvature directions of a surface. Specifically, we use the tree-cotree decomposition of graphs embedded in manifolds, guided with edge weights, in order to produce these cycles. Our algorithm is very quick compared to existing methods, with a worst case running time of O(n log n+gn) where n is the number of faces and g is the surface genus. Further, its flexibility to accommodate different weighting functions and to handle boundaries may be used to produce cycles suitable for a variety of applications and models.