An algorithm that constructsirreducible 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

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

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.

An algorithm that constructs irreducible triangulations of once-punctured surfaces

A

A B

B

Operations on triangulations

Edge shrinking

An algorithm that constructs irreducible triangulations of once-punctured surfaces

Vertex splitting / splitting of a corner

v

u

V1

V2

u

w w

Operations on triangulations

Edge shrinking

An algorithm that constructs irreducible triangulations of once-punctured surfaces

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.

Irreducible triangulations An algorithm that constructs irreducible triangulations of once-punctured surfaces

“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)

The problemAn 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

An algorithm that constructs irreducible triangulations of once-punctured surfaces

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.

Some considerations for the algorithm

An algorithm that constructs irreducible triangulations of once-punctured surfaces

An algorithm that constructs irreducible triangulations of once-punctured surfaces

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.

Sketch of the algorithm

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

An algorithm that constructs irreducible triangulations of once-punctured surfaces

Sketch of the algorithmFirst 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 P

ONE IRREDUCIBLE TRIANGULATION OF S-D J U{P} (Lemma 1 (i))

∂T

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 Ω

i+1(S)

Third step: For each Tє Ω

i+1(S), analyze the cable subgraph of T.

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 store T in Ωi+1

(S) Remove each face sharing e from T

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 store T in Ωi+1

(S) Remove each face sharing e from T TWO IRREDUCIBLE TRIANGULATIONS OF S-D

J U {P,P'}

(Lemma 1 (iii))

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 store T in Ωi+1

(S) Remove that face t from T

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 store T in Ωi+1

(S) Remove that face t from T ONE IRREDUCIBLE TRIANGULATION OF S-D

J U{P}

(Lemma 1 (iii))

∂T

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) store T in Ω

i+1(S)

Remove that vertex v from T

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) store T in Ω

i+1(S)

Remove that vertex v from T ONE IRREDUCIBLE TRIANGULATION OF S-D

(Lemma 1 (ii))

J U {P}

∂T

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

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.

(Lemma 1 (iv))

J U {P}

∂T

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

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 Ω

i+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 withpylonic vertices

An algorithm that constructs irreducible triangulations of once-punctured surfaces

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).

The validity of this procedure

An algorithm that constructs irreducible triangulations of once-punctured surfaces

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, Colin de Verdière & Nakamoto, 2012)

The set of irreducible triangulations of S - D is finite.

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 finiteness of this procedure Incident with all cables of T

The algorithm ENDS

An algorithm that constructs irreducible triangulations of once-punctured surfaces

Input: Ξ0= 21 irreducible triangulations of the torusFirst step: Generate Ξ1 U Ξ2 Second step:

Ξ1 has 433 non-isomorphic: 232 have no pylonic vertex, 193 have an only pylonic vertex

8 have two pylonic vertices.

Ξ2 has 11612 non-isomorphic: none of them is a pylonic triangulation.

(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.

Example: the once-punctured torus

(Mathematica)

(Nauty and gtools)

(Nauty and gtools)

An algorithm that constructs irreducible triangulations of once-punctured surfaces

BOULCH- DE VERDIERE- NAKAMOTO's bounds:

EXAMPLES

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.

There exists a natural number K such that no pylonic triangulation appears after a sequence of K splittings in any irreducible triangulation of S.

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.

An algorithm that constructs irreducible triangulations of once-punctured surfaces

M. J. Chávez, S. Lawrencenko, J. R. Portillo and M. T. Villar - 15 EGC - Sevilla, 2013

¡GRACIAS!