shortest-noncontractible - Computational Topology (Jeff...

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Computational Topology (Jeff Erickson) Shortest Noncontractible Cycles A tie is a noose, and inverted though it is, it will hang a man nonetheless if he’s not careful. — Yann Martel, Life of Pi (2001) (written:) Yakka foob mog. Grug pubbawup zink wattoom gazork. Chumble ¨spuzz. (spoken:) I love loopholes. — Calvin, explaining Newton’s First Law of Motion ‘in his own words’ Bill Watterson, Calvin and Hobbes (January 9, 1995) 8 Shortest Noncontractible Cycles In this lecture, I’ll describe several algorithm to compute the shortest noncontractible cycle in a combi- natorial 2-manifold. The input is a cellularly embedded graph G on a 2-manifold Σ (or equivalently, a polygonal schema for Σ ) with non-negatively weighted edges, and our goal is to compute a cycle in G of minimum length that is not contractible in Σ . This problem has several natural motivations. the length of the shortest non-contractible cycle. In topological graph theory, the length of the shortest non-contractible cycle in an embedded graph is called the handle girth [ 1 ] or edge-width of the embedding [ 15 , 18 ] ; graphs that have embeddings with large edge-width share many useful combinatorial properties with planar graphs. A closely related concept is the face-width or representativity of a graph embedding, which is the minimum number of faces of G intersecting a noncontractible cycle on the surface, or equivalently, half the edge-width of the radial graph G ± [ 17 ] . The length of the shortest noncontractible cycle in a Riemannian manifold is called the systole (to be more specific, the homotopy 1-systole ) of the manifold [ 2 ] . Shortest non-contractible cycles are good indicators of topological noise of geometric surface models reconstructed from point clouds or volume data. The algorithms described below all require that edge weights are non-negative. We treat the graph as a continuous metric space, where the edge weights represent distance. I will assume throughout the lecture that the shortest path between any two vertices of the graph is unique; this assumption can be enforced if necessary using standard perturbation schemes. However, we do not assume that the edge weights satisfy the triangle inequality. 1 8.1 Thomassen’s 3-Path Property The first efficient algorithm to compute shortest non-contractible cycles is an easy consequence of the following observation of Thomassen [ 18 ] . Lemma 8.1. Let x and y be two points on a surface Σ , and let α , β , γ be paths in Σ from x to y . If the loops α · ¯ β and β · ¯ γ are contractible, then the loop α · ¯ γ is also contractible. Proof: The concatenation of any two contractible loops is contractible. ± Corollary 8.2.
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shortest-noncontractible - Computational Topology (Jeff...

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