Optimal System of Loops on an Orientable Surface

Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a fundamental system of loops for M. The resulting disk is a polygon in which the edges are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is triangulated, and that each edge has a given length, we are interested in a shortest (or optimal) system homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give a polynomial (under some reasonable assumptions) algorithm to build such a system. As a byproduct, we get a polynomial algorithm to compute a shortest simple loop homotopic to a given simple loop.