A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with precisely one boundary component.Plan Andaluz de Investigación (Junta de Andalucía

Chávez de Diego, María José

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia (Coordinador)

Lawrecenko, Serge

Márquez Pérez, Alberto (Coordinador)

Portillo Fernández, José Ramón

Urrutia Galicia, Jorge (Coordinador)

Villar Liñán, María Trinidad

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013An algorithm that constructs irreducible triangulations ofonce-punctured surfacesM. J. Chávez∗1, S. Lawrencenko†2, J. R. Portillo‡1, and M. T. Villar§11Universidad de Sevilla, Spain2Russian State University of Tourism and Service, Lyubertsy, Moscow Region, RussiaAbstractA triangulation of a surface is irreducible if there isno edge whose contraction produces another triangu-lation of the surface. In this work we propose an al-gorithm that constructs the set of irreducible trian-gulations of any surface with precisely one boundarycomponent.Introduction and terminologyWe restrict our attention to the following objects:• S = Sg orNk is the closed orientable surface Sg ofgenus g or closed nonorientable surface of nonori-entable genus k. In particular, S0 and S1 are thesphere and torus, N1 and N2 are the projectiveplane and the Klein bottle (respectively).• S − D is S minus an open disk D (the hole).This compact surface is called the once-puncturedsurface. We assume that the boundary ∂S of S,which is equal to ∂D, is homeomorphic to a circle.We use the notation Σ whenever we assume the gen-eral case - that is, Σ ∈ {S, S −D}.If a (finite, undirected, simple) graph G is 2-cellembedded in Σ, the components of Σ − G are calledfaces. A triangulation of Σ with graph G is a 2-cellembedding T : G→ Σ in which each face is boundedby a 3-cycle (that is, a cycle of length 3) of G and anytwo faces are either disjoint, share a single vertex,or share a single edge. We denote by V = V (T ),E = E(T ), and F = F (T ) the sets of the vertices, theedges, and the faces of T , respectively. Equivalently,the triangulation T of a surface can be defined as ahypergraph of rank 3 or a 3-graph, with the vertexset V (T ) and the collection F (T ) of triplets of V (T )(called 3-edges, or triangles, of T ) (see [4]).By G(T ) we denote the graph (V (T ), E(T )) of tri-angulation T . Two triangulations T1 and T2 are called∗Email: mjchavez@us.es. Research supported by PAI FQM-189†Email: lawrencenko@hotmail.com.‡Email: josera@us.es. Research supported by PAI FQM-164§Email: villar@us.es. Research supported by PAI FQM-164isomorphic, denoted T1 ∼= T2, if there is a bijectionα : V (T1)→ V (T2) such that uvw ∈ F (T1) if and onlyif α(u)α(v)α(w) ∈ F (T2). Throughout this paper wedistinguish triangulations only up to isomorphism.In the case Σ = S − D let ∂T , which is equal to∂D, denote the boundary cycle of T . The vertices andedges of ∂T are called boundary vertices and boundaryedges of T , respectively.A triangulation of a punctured surface is irreducible(term which is more accurately defined in Section 1)if no edge can be shrunk without producing multipleedges or changing the topological type of the surface.The irreducible triangulations of Σ form a basis forthe family of all triangulations of Σ, in the sense thatany triangulation of Σ can be obtained from a memberof the basis by applying the splitting operation (intro-duced in Section 1) a finite number of times. To havesuch a basis in hand can be very useful in practicalapplication of triangulation generating; the papers [6]and [19] are worth mentioning.It is known that for every surface Σ the basis ofirreducible triangulations is finite (the case of closedsurfaces is proved in [2], [13], [12], and [7] and thecase of surfaces with boundary is proved in [3]). Atpresent such bases are known only for seven closedsurfaces and two once-punctured surfaces: the sphere([14]), projective plane ([1]), torus ([8]), Klein bottle([9] and [15]), S2, N3, and N4 ([16, 17]), the disk andMöbius band ([5]).In this paper we present an algorithm which isdesigned as an application of some recent advanceson the study of irreducible triangulations of once-punctured surface collected in [5]. Specifically, Lem-mas 1-3 (in Section 1) are the main supporting the-oretical results for the mentioned algorithm. As aparticular example, all the non-isomorphic combina-torial types (293 in number) of triangulations of theonce-punctured torus are determined.1 Previous resultsLet T be a triangulation of Σ. An unordered pairof distinct adjacent edges vu and vw of T is called43An algorithm that constructs irreducible triangulations of once-punctured surfacesa corner of T at vertex v, denoted by 〈u, v, w〉(=〈w, v, u〉). The splitting of a corner 〈u, v, w〉, de-noted by sp〈u, v, w〉, is the operation which consistsof cutting T open along the edges vu and vw andthen closing the resulting hole with two new triangu-lar faces, v′v′′u and v′v′′w, where v′ and v′′ denotethe two images of vertex v appearing as a result ofcutting. Under this operation, vertex v is extendedto the edge v′v′′ and the two faces having this edge incommon are inserted into the triangulation; thereforethe order increases by one and the number of edgesincreases by three.If a corner 〈u, v, w〉 is composed of two edges vuand vw neighboring around vertex v, sp〈u, v, w〉 isequivalent to the stellar subdivision of the face uvw.Especially in the case {Σ = S − D, uv ∈E(T ), and v ∈ V (T )}, the operation sp〈u, v] of split-ting a truncated corner 〈u, v] produces a single trian-gular face uv′v′′, where v′v′′ ∈ E(∂(sp〈u, v](T ))).Under the inverse operation, shrinking the edgev′v′′, denoted by sh〉v′v′′〈, this edge collapses to asingle vertex v, the faces v′v′′u and v′v′′w collapseto the edges vu and vw, respectively. Thereforesp〈u, v, w〉 ◦ sh〉v′v′′〈(T ) = T . It should be noticedthat in the case {Σ = S − D, v′v′′ ∈ E(∂T )}, thereis only one face incident with v′v′′, and only that sin-gle face collapses to an edge under sh〉v′v′′〈. Clearly,the operation of splitting doesn’t change the topolog-ical type of Σ if Σ ∈ {S, S − D}. We demand thatthe shrinking operation must preserve the topologicaltype of Σ as well; moreover, multiple edges must notbe created in a triangulation. A 3-cycle of T is callednonfacial if it doesn’t bound a face of T . In the casein which an edge e ∈ E(T ) occurs in some nonfacial3-cycle, if we still insist on shrinking e, multiple edgeswould be produced, which would expel sh〉e〈(T ) fromthe class of triangulations. An edge e is called shrink-able or a cable if sh〉e〈(T ) is still a triangulation ofΣ; otherwise the edge is called unshrinkable or a rod.The subgraph of G(T ) made up of all cables is calledthe cable-subgraph of G(T ).The impediments to edge shrinkability in a trian-gulation T of a punctured surface S−D are identifiedin [2, 3, 1, 8]; an edge e ∈ E(T ) is a rod if and only ife satisfies one of the following conditions:(1) e is in a nonfacial 3-cycle of G(T ). In particular,e is a boundary edge in the case in which the boundarycycle is a 3-cycle.(2) e is a chord of D -that is, the end vertices of eare in V (∂D) but e /∈ E(∂D).From now on, we assume that S 6= S0 and makean agreement that by “non-facial 3-cycle” we mean anon-null-homotopic 3-cycle whenever we refer to con-ditions (1) and (2). Therefore, an edge e is a rod pro-vided e occurs in some non-null-homotopic 3-cycle,and e is a cable otherwise. Especially, the boundaryedges of stellar subdivided faces are now regarded ascables unless they occur in some non-null-homotopic3-cycles. The convenience of this agreement is thatonce a rod turns into a cable in the course of anysplitting sequence, it always remains a cable underforthcoming splittings.A triangulation is said to be irreducible if it is freeof cables or equivalently, each edge is a rod. For in-stance, a single triangle is the only irreducible trian-gulation of the disk S0 −D.Let T be an irreducible triangulation of a puncturedsurface S −D where S 6= S0. Let us close the hole inT by restoring the disk D add a vertex, p, in D, andjoin p to the vertices in ∂D. We thus obtain a triangu-lation, T ∗, of the closed surface S. In this setting wecall D the patch, call p the central vertex of the patch,and say that T is obtained from the correspondingtriangulation T ∗ of S by the patch removal.Notice that T ∗ may be an irreducible triangulationof S but not necessarily. Using the assumption thatT is irreducible and the fact that each cable of T ∗fails to satisfy condition (1) (in the strong non-null-homotopic sense), it can be easily seen that in thecase T ∗ is not irreducible, all cables of T ∗ have to beentirely in D ∪ ∂D and, moreover, there is no cablewhich is entirely in ∂D whenever the length of theboundary cycle ∂D is greater than or equal to 4. Inparticular, we observe that each chord of D (if any) isa rod in T because it meets condition (2), and is alsoa rod in T because it meets condition (1). We nowcome to a lemma which is to be stated shortly aftersome necessary definitions.A vertex of a triangulation R of S is called a pylonicvertex if that vertex is incident with all cables of R. Atriangulation which has at least one cable and at leastone pylonic vertex is called a pylonic triangulation.A triangulation may have a unique cable and there-fore two pylonic vertices. However, if the number ofcables in a pylonic triangulation R is at least 2, R hasexactly one pylonic vertex.Lemma 1 Suppose an irreducible triangulation T ofa punctured surface S −D (S 6= S0) is obtained fromthe corresponding triangulation T ∗ of S by the patchremoval. If T ∗ has at least two cables, then either thecentral vertex p of the patch is the only pylonic vertexof T ∗, or else the length of ∂D is equal to 3.Let Ξn = Ξn(S) denote the set of triangulations ofa fixed closed surface S that can be obtained from anirreducible triangulation of S by a sequence of exactlyn repeated splittings.In the following result, by the “removal of a vertexv” we mean the removal of v together with the inte-riors of the edges and faces incident with v and bythe “removal of a face” we mean the removal of theinterior of that face.44XV Spanish Meeting on Computational Geometry, June 26-28, 2013Lemma 2 Each irreducible triangulation T of S−D(S 6= S0) can be obtained either(i) by removing a vertex from a triangulation inΞ0 = Ξ0(S), or(ii) by removing a pylonic vertex from a triangulationin Ξ1 ∪ Ξ2 ∪ · · · ∪ ΞK , where the constant K isprovided by [3] (whenever a pylonic triangulationoccurs), or(iii) by removing either of the two faces containinga cable in their boundary 3-cycles provided thatcable is unique in a triangulation in Ξ1 (wheneversuch a situation occurs), or(iv) by removing the face containing two, or three, ca-bles in its boundary 3-cycle provided those two,or three, cables collectively form the whole cable-subgraph in a triangulation in Ξ1 ∪ Ξ2 (if such asituation occurs).Lemma 3 If a triangulation of S has at least two ca-bles but has no pylonic vertex, then no pylonic vertexcan be created under further splitting of the triangu-lation.2 Sketch of the algorithmIn this section triangulations are considered to behypergraphs of rank 3 or 3-graphs. Let T be a 3-graph with V = V (T ) and F = F (T ) the setsof the vertices and the triangles of T . This 3-graph can be uniquely represented as a bipartitegraph BT = (V (BT ), E(BT )) in the following way:V (BT ) = V (T )∪F (T ), uv ∈ E(BT ) if and only if thevertex u lies in the triangle v in T .The algorithm input is the set I of irreducible tri-angulations of a closed surface S 6= S0. The outputof the algorithm is the set of all non-isomorphic com-binatorial types of irreducible triangulations of theonce-punctured surface S −D.The first step is the generation of the set Ξ1 ∪ Ξ2from the set I. Next, every 3-graph T ∈ Ξ1 ∪ Ξ2is then represented by their corresponding bipartitegraph BT . This has been implemented with the com-putational package Mathematica ([18]).The second step consists of discarding all duplicatebipartite graphs and then, all duplicate triangulationsin Ξ1∪Ξ2 will be discarded. That is, the obtention ofall non-isomorphic triangulations, denoted Ξ̃1∪ Ξ̃2 re-spectively, which is implemented with the computingpackages Nauty (and gtools, [10], [11]).Next, all pylonic vertices in Ξ̃1∪Ξ̃2 are detected andoperations (i)–(iv) described in Lemma 2 are appliedto obtain irreducible triangulations (using Mathemat-ica).If Ξ2 has no pylonic triangulation, immediately pro-ceed to the final step: Discard all duplicate triangula-tions by using Nauty and gtools. Otherwise generateΞ3 and apply the preceding steps to Ξ3. Repeat thisprocedure to process Ξ4,Ξ5, . . . until no pylonic tri-angulation is left in the current Ξn; then the processterminates and a required output is produced.The validity of this procedure is justified by Lem-mas 1 - 3 along with the results of [3]. In particular,the finiteness of the procedure is deduced from theupper bound [3] on the number of vertices in an irre-ducible triangulation of S−D. In particular, that up-per bound implies (along with Lemma 3) that Ξn doesnot have a pylonic triangulation for any n ≥ K + 1,where K = 945 for S1 and K = 376 for N1. In realityK is much less than these values. By computer ver-ification (and also by hand) we have checked that infact K = 1 for S1, and K = 2 for N1.Let us now mention two examples.Firstly, this algorithm has been implemented for theset of two irreducible triangulations of N1 ([1]). Thealgorithm gives a set of 6 irreducible triangulations ofthe Möbius band, N1−D, which is precisely the sameas that obtained by some of the authors of this workin [5], although by using no computational package.Secondly, we introduce the details of the torus case,S1.Example: the once-punctured torusInput: The set of 21 irreducible triangulations of S1(as they are labelled in [8]).• Ξ1(S1) has 433 non-isomorphic triangulations:232 of them have no pylonic vertex, 193 havean only pylonic vertex and 8 have two pylonicvertices.• Ξ2(S1) has 11612 non-isomorphic triangulations,none of them is a pylonic triangulation.• Operations described in Lemma 2 provide:(i) 184 triangulations; only 80 of them are non-isomorphic.(ii) 209 triangulations; only 203 of them arenon-isomorphic.(iii) 16 triangulations, 10 of them are non-isomorphic.(iv) 0 triangulations.Output: 293 non-isomorphic combinatorial types ofirreducible triangulations of the once-punctured torusS1 −D.3 Final conclusionIt is clear that this algorithm can be implemented forany closed surface whenever its basis of irreducible45An algorithm that constructs irreducible triangulations of once-punctured surfacestriangulations is known. In a future contribution wehope to present the set of irreducible triangulations ofthe once-punctured Klein bottle, N2 −D.References[1] D. Barnette (D. W. Barnette). Generating the trian-gulations of the projective plane. J. Comb. Theory,Ser. B, 33, (1982), 222–230.[2] D. W. Barnette, A. L. Edelson. All 2-manifolds havefinitely many minimal triangulations. Isr. J. Math.,67, (1989), No. 1:123–128.[3] A. Boulch, É. Colin de Verdière, A. Nakamoto. Ir-reducible triangulations of surfaces with boundary.Graphs and Combinatorics DOI 10.1007/s00373-012-1244-1, 2012.[4] J. Bracho, R. Strausz, Nonisomorphic complete tri-angulations of a surface. Discrete Math. 232, (2001),11–18.[5] M. J. Chávez, S. A. Lawrencenko, A. Quintero, M. T.Villar. Irreducible triangulations of the Möbius band.Preprint 2013.[6] L. Giomi, M. J. Bowick. Elastic theory of defects intoroidal crystals. Eur. Phys. J. E., 27, (2008), 275–296.[7] G. Joret, D. R. Wood. Irreducible triangulations aresmall. J. Comb. Theory, Ser. 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Comb. Theory, Ser. B, 96, No.6, (2006), 964–972.[16] T. Sulanke. Generating irreducible triangulationsof surfaces, preprint, 2006, arXiv:math/0606687v1[math.CO].[17] T. Sulanke. Irreducible triangulations of low genussurfaces, preprint, 2006, arXiv:math/0606690v1[math.CO].[18] Wolfram Research, Inc. Mathematica Version 8.0(Wolfram Research, Inc.) Champaign, Illinois 2010.[19] Y. Zhongwey, J. Shouwei. STL file generation fromdigitised data points based on triangulation of 3Dparametric surfaces.Int. J. Adv. Manuf. Technol. 23,(2004), 882–888.46

An algorithm that constructs irreducible triangulations of once-punctured surfaces

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An algorithm that constructs irreducible triangulations of once-punctured surfaces

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