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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