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Nov. 5th, 2013
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Nov. 5th, 2013 08:17 amI cannot solve a simple combinatorial problem, can anyone help?
Let T be a finite triangulation of a compact 2d manifold. I am going to describe an equivalence relation on vertices of T. An elementary equivalence relation is associated with any edge E: E is shared by two triangles A and B, and we declare the vertices of A and B which do not belong to E equivalent. Two vertices are equivalent if they are related by a chain of elementary equivalence relations. The equivalence relation is entirely determined by T. In general, there is more than one equivalence class of vertices.
Now, is it always possible to subdivide T so that all vertices of the new triangulation T' are equivalent?
Let T be a finite triangulation of a compact 2d manifold. I am going to describe an equivalence relation on vertices of T. An elementary equivalence relation is associated with any edge E: E is shared by two triangles A and B, and we declare the vertices of A and B which do not belong to E equivalent. Two vertices are equivalent if they are related by a chain of elementary equivalence relations. The equivalence relation is entirely determined by T. In general, there is more than one equivalence class of vertices.
Now, is it always possible to subdivide T so that all vertices of the new triangulation T' are equivalent?