0
Upcoming SlideShare
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Standard text messaging rates apply

# Triangulaciones irreducibles en el toro perforado

87

Published on

Talk in conference

Talk in conference

0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
87
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
1
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Transcript

• 1. An algorithm that constructs irreducible triangulations of once-punctured surfaces M. J. Ch&#xE1;vez, 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 &#x2013; D and &#x2202;D is the boundary of S &#x2013; D. The disk is the punctured sphere The M&#xF6;bius band is the punctured projective plane
• 3. An algorithm that constructs irreducible triangulations of once-punctured surfaces Preliminaries The M&#xF6;bius 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, &#x2202;D = &#x2202;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 &#x201C;The tetrahedron is the only irreducible triangulation for the sphere&#x201D;. (Steinitz, 1934) &#x201C;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&#x201D;. (Barnette, 1989, Nakamoto &amp; Ota, 1995, Negami, 2001, Joret &amp; Wood, 2010) Proyective plane, (Barnette 1982) Torus, (Lawrencenko 1987 ) Klein Bottle, (Lawrencenko-Negami 1997, Sulanke 2005) Double Torus, N3 , N4 (Sulanke, 2006) By computing! &#x201C;For any surface with boundary S, the set of irreducible triangulation is finite&#x201D;. (Boulch, Colin de Verdi&#xE8;re &amp; Nakamoto, 2012) M&#xF6;bius band, (Ch&#xE1;vez, Lawrencenko, Quintero &amp; 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 &#x454; E(BT ) if and only if the vertex u lies in the triangle v &#x454; 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 (&#x2260; 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 =&#x39E;0(S) (the set of irreducible triangulations of the closed surface S.) J = &#xD8; (set of irreducible triangulations of the once-punctured surface S-D.) For each T&#x454; &#x39E;0 (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 =&#x39E;0(S) (the set of irreducible triangulations of the closed surface S.) &#xA0; J = &#xD8; (the set of irreducible triangulations of the once-punctured surface S-D.) For each T&#x454; &#x39E;0 (S) and each vertex v in T remove v from T ONE IRREDUCIBLE TRIANGULATION OF S-D &#x2202;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 T&#x454; &#x39E;i (S), split every vertex of T and generate &#x39E;i+1(S). Discard all duplicate (=combinatorially isomorphic) triangulations of &#x39E;i+1 (S) by using the bipartite graph &#x3A9;i+1(S) Third step: For each T&#x454; &#x3A9;i+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 T&#x454; &#x3A9;i+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 &#x3A9;i+1(S)
• 15. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For i and for each T&#x454; &#x3A9;i+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 &#x3A9;i+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 T&#x454; &#x3A9;i+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 &#x3A9;i+1(S)
• 17. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each T&#x454; &#x3A9;i+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 &#x3A9;i+1(S) ONE IRREDUCIBLE TRIANGULATION OF S-D J U{P} (Lemma 1 (iii)) &#x2202;T
• 18. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each T&#x454; &#x3A9;i+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 &#x3A9;i+1(S)
• 19. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each T&#x454; &#x3A9;i+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 &#x3A9;i+1(S) ONE IRREDUCIBLE TRIANGULATION OF S-D J U {P} &#x2202;T (Lemma 1 (ii))
• 20. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each T&#x454; &#x3A9;i+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 &#x3A9;i+1(S)
• 21. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each T&#x454; &#x3A9;i+1(S) analyze the cable subgraph of T. CASE D: Three cables defining a face t in T discard T in &#x3A9;i+1(S) Remove that face t from T ONE IRREDUCIBLE TRIANGULATION OF S-D. J U {P} (Lemma 1 (iv)) &#x2202;T
• 22. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each T&#x454; &#x3A9;i+1(S) analyze the cable subgraph of T. CASE E: Otherwise discard T from &#x3A9;i+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 T&#x454; &#x3A9;i+1(S) analyze the cable subgraph of T. Apply Lemma 1 (ii)-(iv) (according to cases A to E). Discard all duplicate triangulations in &#x3A9;i+1(S). While &#x3A9;i+1(S) &#x2260; &#xD8; 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 &#x2260; sphere) can be obtained either (I) by removing a vertex from a triangulation in &#x39E;0(S), or (II) by removing a pylonic vertex from a pylonic triangulation in &#x39E;1 U &#x39E;2 U&#x2026;U &#x39E;K , 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 &#x39E;1 (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 &#x39E;1 U &#x39E;2 (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. (Ch&#xE1;vez, Lawrencenko, Quintero &amp; Villar, 2013) The set of irreducible triangulations of S - D is finite. (Boulch, Colin de Verdi&#xE8;re &amp; 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: &#x39E;0= 21 irreducible triangulations of the torus First step: Generate &#x39E;1 U &#x39E;2 Second step: &#x39E;1 has 433 non-isomorphic: 232 have no pylonic vertex, 193 have an only pylonic vertex (Nauty and gtools) 8 have two pylonic vertices. (Mathematica) &#x39E;2 has 11612 non-isomorphic: none of them is a pylonic triangulation. (Nauty and gtools) (I) Removing a vertex from &#x39E;0 : 184 triangulations. 80 are non-isomorphic. (II) Removing a pylonic vertex from &#x39E;1U&#x39E;2 : 209 triangulations. 203 are non-isomorphic. (III) Removing faces from triangulation with a unique cable in &#x39E;1:16 triangulations. 10 are non-isomorphic. (IV) No face is bounded by two cables in &#x39E;1U&#x39E;2 :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 &#xA1;GRACIAS! M. J. Ch&#xE1;vez, S. Lawrencenko, J. R. Portillo and M. T. Villar - 15 EGC - Sevilla, 2013