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Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
Triangulaciones irreducibles en el toro perforado
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Triangulaciones irreducibles en el toro perforado

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  • 1. An algorithm that constructs irreducible triangulations of once-punctured surfaces M. J. Chávez, J. R. Portillo, M. T. Villar Universidad de Sevilla and S. Lawrencenko Russian State University of Tourism and Service 15 EGC - Sevilla, 2013
  • 2. An algorithm that constructs irreducible triangulations of once-punctured surfaces Preliminaries A once-punctured surface is a compact surface with a hole obtained from a closed compact connected (orientable or non-orientable) surface S by the deletion of the interior of a disk (hole). It is denoted S – D and ∂D is the boundary of S – D. The disk is the punctured sphere The Möbius band is the punctured projective plane
  • 3. An algorithm that constructs irreducible triangulations of once-punctured surfaces Preliminaries The Möbius band is the punctured projective plane A triangulation T on a surface S is a simple graph T embedded in S so that each face is bounded by a 3-cycle and any two faces share at most one edge. In case that S is a once-punctured surface, ∂D = ∂T denotes the boundary cycle of T. A B B A
  • 4. An algorithm that constructs irreducible triangulations of once-punctured surfaces Operations on triangulations Edge shrinking
  • 5. An algorithm that constructs irreducible triangulations of once-punctured surfaces Operations on triangulations Vertex splitting / splitting of a corner u u V1 v V2 w w Edge shrinking
  • 6. An algorithm that constructs irreducible triangulations of once-punctured surfaces Irreducible triangulations T is a triangulation of a surface S. An edge e of T is shrinkable or a cable if the graph obtained after shrinking e, is still a triangulation of S. T is said to be irreducible if it is free of cables.
  • 7. An algorithm that constructs irreducible triangulations of once-punctured surfaces The problem “The tetrahedron is the only irreducible triangulation for the sphere”. (Steinitz, 1934) “For any closed surface S, there is a finite set of irreducible triangulations of S, I, so that any other triangulation of S can be obtained from a triangulation of I by applying a sequence of vertex splitting”. (Barnette, 1989, Nakamoto & Ota, 1995, Negami, 2001, Joret & Wood, 2010) Proyective plane, (Barnette 1982) Torus, (Lawrencenko 1987 ) Klein Bottle, (Lawrencenko-Negami 1997, Sulanke 2005) Double Torus, N3 , N4 (Sulanke, 2006) By computing! “For any surface with boundary S, the set of irreducible triangulation is finite”. (Boulch, Colin de Verdière & Nakamoto, 2012) Möbius band, (Chávez, Lawrencenko, Quintero & Villar, 2013)
  • 8. An algorithm that constructs irreducible triangulations of once-punctured surfaces The problem: once-punctured surfaces If the set of irreducible triangulations of S is known The set of irreducible triangulations of the once-punctured surface S-D is known
  • 9. An algorithm that constructs irreducible triangulations of once-punctured surfaces Some considerations for the algorithm Any triangulation T is considered to be a hypergraph of rank 3 or 3-graph. T is determined by its vertex set V=V(T) and its triangle set F=F(T) T can be uniquely represented as a bipartite graph BT=(V(BT ), E(BT )) V(BT )=V(T)U F(T) uv є E(BT ) if and only if the vertex u lies in the triangle v є T. Two triangulations T and T' are combinatorially isomorphic if and only if their bipartite graphs BT and BT' are isomorphic.
  • 10. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Input : the set I of irreducible triangulations of a closed surface S (≠ sphere). Output: the set of all non-isomorphic combinatorial types of irreducible triangulations of the once-punctured surface S-D.
  • 11. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm First step: For i=0 consider I =Ξ0(S) (the set of irreducible triangulations of the closed surface S.) J = Ø (set of irreducible triangulations of the once-punctured surface S-D.) For each Tє Ξ0 (S) and each vertex v in T remove v from T P
  • 12. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm First step: For i=0 consider I =Ξ0(S) (the set of irreducible triangulations of the closed surface S.)   J = Ø (the set of irreducible triangulations of the once-punctured surface S-D.) For each Tє Ξ0 (S) and each vertex v in T remove v from T ONE IRREDUCIBLE TRIANGULATION OF S-D ∂T P J U{P} (Lemma 1 (i))
  • 13. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Second step: For i and for each Tє Ξi (S), split every vertex of T and generate Ξi+1(S). Discard all duplicate (=combinatorially isomorphic) triangulations of Ξi+1 (S) by using the bipartite graph Ωi+1(S) Third step: For each Tє Ωi+1(S), analyze the cable subgraph of T.
  • 14. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For i and for each Tє Ωi+1(S) analyze the cable subgraph of T. CASE A: Only one cable e in T Remove each face sharing e from T store T in Ωi+1(S)
  • 15. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For i and for each Tє Ωi+1(S) analyze the cable subgraph of T. CASE A: Only one cable e in T Remove each face sharing e from T store T in Ωi+1(S) TWO IRREDUCIBLE TRIANGULATIONS OF S-D J U {P,P'} (Lemma 1 (iii))
  • 16. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE B: Two cables e, e' share a face t in T Remove that face t from T store T in Ωi+1(S)
  • 17. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE B: Two cables e, e' share a face t in T Remove that face t from T store T in Ωi+1(S) ONE IRREDUCIBLE TRIANGULATION OF S-D J U{P} (Lemma 1 (iii)) ∂T
  • 18. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE C: Two or more cables incident in a vertex v in T (but not in case B) Remove that vertex v from T store T in Ωi+1(S)
  • 19. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE C: Two or more cables incident in a vertex v in T (but not in case B) Remove that vertex v from T store T in Ωi+1(S) ONE IRREDUCIBLE TRIANGULATION OF S-D J U {P} ∂T (Lemma 1 (ii))
  • 20. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE D: Three cables defining a face t in T Remove that face t from T discard T in Ωi+1(S)
  • 21. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE D: Three cables defining a face t in T discard T in Ωi+1(S) Remove that face t from T ONE IRREDUCIBLE TRIANGULATION OF S-D. J U {P} (Lemma 1 (iv)) ∂T
  • 22. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE E: Otherwise discard T from Ωi+1(S) NO IRREDUCIBLE TRIANGULATION OF S-D is obtained from T. Lemma If a triangulation T of S has at least two cables but has no pylonic vertex, then no pylonic vertex can be created under further splitting of the triangulation. Incident with all cables of T
  • 23. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. Apply Lemma 1 (ii)-(iv) (according to cases A to E). Discard all duplicate triangulations in Ωi+1(S). While Ωi+1(S) ≠ Ø do i+1 and go to Second step Else go to Final step Final step: Discard all duplicate triangulations in J END Triangulations with pylonic vertices
  • 24. An algorithm that constructs irreducible triangulations of once-punctured surfaces The validity of this procedure LEMMA 1 Each irreducible triangulation T of S-D (S ≠ sphere) can be obtained either (I) by removing a vertex from a triangulation in Ξ0(S), or (II) by removing a pylonic vertex from a pylonic triangulation in Ξ1 U Ξ2 U…U ΞK , where K is provided by BOULCH-DE VERDIERE- NAKAMOTO's result. (III) by removing either of the two faces containing a cable in their boundary 3-cycles provided that cable is unique in a triangulation in Ξ1 (whenever such a situation occurs), or (IV) by removing the face containing two, or three, cables in its boundary 3-cycle provided those two, or three, cables collectively form the whole cable-subgraph in a triangulation in Ξ1 U Ξ2 (if such a situation occurs).
  • 25. An algorithm that constructs irreducible triangulations of once-punctured surfaces Incident with all cables of T The finiteness of this procedure Removing a pylonic vertex in a triangulation of a closed surface S gives rise to an irreducible triangulation of S - D. (Chávez, Lawrencenko, Quintero & Villar, 2013) The set of irreducible triangulations of S - D is finite. (Boulch, Colin de Verdière & Nakamoto, 2012) There exists a natural number K such that no pylonic triangulation appears after a sequence of K splittings in any irreducible triangulation of S. The algorithm ENDS
  • 26. An algorithm that constructs irreducible triangulations of once-punctured surfaces Example: the once-punctured torus Input: Ξ0= 21 irreducible triangulations of the torus First step: Generate Ξ1 U Ξ2 Second step: Ξ1 has 433 non-isomorphic: 232 have no pylonic vertex, 193 have an only pylonic vertex (Nauty and gtools) 8 have two pylonic vertices. (Mathematica) Ξ2 has 11612 non-isomorphic: none of them is a pylonic triangulation. (Nauty and gtools) (I) Removing a vertex from Ξ0 : 184 triangulations. 80 are non-isomorphic. (II) Removing a pylonic vertex from Ξ1UΞ2 : 209 triangulations. 203 are non-isomorphic. (III) Removing faces from triangulation with a unique cable in Ξ1:16 triangulations. 10 are non-isomorphic. (IV) No face is bounded by two cables in Ξ1UΞ2 :0 triangulations. Output: The list of 203 + 80 + 10 = 293 non-isomorphic combinatorial types of irreducible triangulations of the once-punctured torus.
  • 27. An algorithm that constructs irreducible triangulations of once-punctured surfaces EXAMPLES There exists a natural number K such that no pylonic triangulation appears after a sequence of K splittings in any irreducible triangulation of S. BOULCH- DE VERDIERE- NAKAMOTO's bounds: For the torus, K = 945; for the Projective plane, K = 376 By computer verification and also by hand we have checked that, in fact: K = 1 for the torus and K=2 for the Projective plane.
  • 28. An algorithm that constructs irreducible triangulations of once-punctured surfaces Final conclusion This algorithm can be implemented for any closed surface whenever its basis of irreducible triangulations is known. In a future contribution we hope to present the set of irreducible triangulations of the once-punctured Klein bottle.
  • 29. An algorithm that constructs irreducible triangulations of once-punctured surfaces ¡GRACIAS! M. J. Chávez, S. Lawrencenko, J. R. Portillo and M. T. Villar - 15 EGC - Sevilla, 2013

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