In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs with triangular outer face is connected and has diameter O(n2).European Science FoundationAustrian Science FundMinisterio de Ciencia e InnovaciónJunta de Castilla y Leó

Aichholzer, Oswin

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia (Coordinador)

Hackl, Thomas

Márquez Pérez, Alberto (Coordinador)

Orden Martín, David

Pilz, Alexander

Saumell Mendiola, María

Urrutia Galicia, Jorge (Coordinador)

Vogtenhuber, Birgit

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Flips in combinatorial pointed pseudo-triangulationswith face degree at most four (extended abstract)Oswin Aichholzer∗1, Thomas Hackl∗1, David Orden†2, Alexander Pilz∗1, Maria Saumell‡3, andBirgit Vogtenhuber∗11Institute for Software Technology, Graz University of Technology, Austria.2Departamento de Física y Matemáticas, Universidad de Alcalá, Spain.3Département d'Informatique, Université Libre de Bruxelles, Belgium.AbstractIn this paper we consider the ip operation for com-binatorial pointed pseudo-triangulations where faceshave size 3 or 4, so-called combinatorial 4-PPTs. Weshow that every combinatorial 4-PPT is stretchableto a geometric pseudo-triangulation, which in generalis not the case if faces may have size larger than 4.Moreover, we prove that the ip graph of combina-torial 4-PPTs with triangular outer face is connectedand has diameter O(n2).1 IntroductionGiven a graph of a certain class, a ip is the op-eration of removing one edge and inserting a dier-ent one such that the resulting graph is again of thesame class. For the class of maximal planar (sim-ple) graphs, any combinatorial embedding (clockwiseorder of edges around each vertex) has only faces ofsize 3 and hence is called a combinatorial triangu-lation. Flips in combinatorial triangulations removethe common edge of two triangular faces and replaceit by the edge between the two vertices not shared bythe faces, provided that these two vertices where notalready joined by an edge. Combinatorial triangula-tions have a geometric counterpart in triangulationsof point sets in the plane, which are maximal plane∗Email: [oaich|thackl|apilz|bvogt]@ist.tugraz.at. Researchof OA and BV partially supported by the ESF EUROCORESprogramme EuroGIGA  CRP `ComPoSe', Austrian ScienceFund (FWF): I648-N18. Research of TH supported by the Aus-trian Science Fund (FWF): P23629-N18 `Combinatorial Prob-lems on Geometric Graphs'. AP is a recipient of a DOC-fellowship of the Austrian Academy of Sciences.†Email: david.orden@uah.es. Research partially supportedby MICINN Project MTM2011-22792, ESF EUROCORESprogramme EuroGIGA - ComPoSe IP04 - MICINN ProjectEUI-EURC-2011-4306 and Junta de Castilla y León ProjectVA172A12-2.‡Email: maria.saumell.m@gmail.com. Research supportedby ESF EuroGIGA project ComPoSe as F.R.S.-FNRS - EU-ROGIGA NR 13604 and by ESF EuroGIGA project GraDR asGAR GIG/11/E023.geometric (straight-line) graphs with predened ver-tex positions. In this geometric setting there is also aip operation, for which a dierent restriction applies:An edge can be ipped if and only if the two adjacenttriangles form a convex quadrilateral (otherwise thenew edge would create a crossing).Flips in (combinatorial) triangulations have beenthoroughly studied. See [4] for a survey. A promi-nent question about ips is to study the ip graph.This is an abstract graph whose vertices are the mem-bers of the same graph class having the same numberof vertices, and in which two graphs are neighborsif and only if one can be transformed into the otherby a single ip. For both, combinatorial triangula-tions and triangulations (with xed vertex positions),the ip graph is connected. However, the dierentsettings imply linear and quadratic diameter, respec-tively (see [4] for references).Triangulations have a natural generalization inpseudo-triangulations. They have become a popu-lar structure in Computational Geometry within thelast two decades, with applications in, e.g., rigid-ity theory and motion planning. See [7] for a sur-vey. A pseudo-triangle is a simple polygon in theplane with exactly three convex vertices (i.e., verticeswhose interior angle is smaller than π). A pseudo-triangulation T of a nite point set S in the planeis a partition of the convex hull of S into pseudo-triangles such that the union of the vertices of thepseudo-triangles is exactly S. Triangulations are aparticular type of pseudo-triangulations, actually theones with the maximum number of edges. Thosewith the minimum number of edges are the so-calledpointed pseudo-triangulations, in which every vertex ispointed, i.e., incident to a reex angle (an angle largerthan π).Flips can also be dened for the class of pseudo-triangulations of point sets in the plane. The ipgraph for general pseudo-triangulations is known tobe connected, as well as the subgraph induced bypointed pseudo-triangulations. The currently bestknown bound on the diameter is O(n log n) for both131Flips in combinatorial pointed pseudo-triangulations with face degree at most four (extended abstract)ip graphs [2, 3].In a pseudo-triangulation, the pseudo-triangles canhave linear size. Hence, in contrast to triangula-tions, the ip operation can no longer be computedin constant time. This fact led to the consideration ofpseudo-triangulations in which the size of the pseudo-triangles is bounded by a constant. Kettner et al. [5]showed that every point set admits a pointed pseudo-triangulation with face degree at most four (except,maybe, for the outer face). We call such a pseudo-triangulation a 4-PPT.On the one hand, 4-PPTs behave nicely for prob-lems which are hard for general pseudo-triangulations.For instance, they are always properly 3-colorable,while 3-colorability is NP-complete to decide for gen-eral pseudo-triangulations [1]. On the other hand,known properties of general pseudo-triangulations re-main open for 4-PPTs. For instance, it is not knownwhether the ip graph of 4-PPTs is connected, evenfor the basic case of a triangular convex hull.The aim of this paper is to make a step towardsanswering this last question, by considering the com-binatorial counterpart of 4-PPTs.A combinatorial pseudo-triangulation [6] is a topo-logical embedding of a planar simple graph togetherwith an assignment of labels reex/convex to its an-gles such that (1) every interior face has exactly threeangles labeled convex, (2) all the angles of the outerface are labeled reex, and (3) no vertex is incidentto more than one reex angle.Note that this labeling fullls the same proper-ties as actual reex/convex angles in a (geometric)pseudo-triangulation. This analogy with the geomet-ric case goes on by calling pointed vertices in a com-binatorial pseudo-triangulation those which, indeed,are incident to one angle labeled reex. Then, com-binatorial pointed pseudo-triangulations are those inwhich every vertex is pointed. Combinatorial pointedpseudo-triangulations with face degree at most four(except, maybe, for the outer face), will be called com-binatorial 4-PPTs.2 PropertiesLemma 1 Let G be a combinatorial 4-PPT and Hbe a subgraph of G with |V (H)| ≥ 3. Then H has atleast 3 vertices whose reex angle is contained in theouter face of H (corners of rst type in [6]).Corollary 2 In any combinatorial 4-PPT of the in-terior of a simple cycle with b vertices, of which chave the reex angle inside the cycle, the number t oftriangular faces is given by t = b− 2c− 2.A combinatorial pseudo-triangulation has the gen-eralized Laman property if every subset of x non-poin-ted vertices and y pointed vertices, where x+y ≥ 2,induces a subgraph with at most 3x + 2y − 3 edges.Both this property and the number of reex anglesfrom Lemma 1 are related to the stretchability of acombinatorial pseudo-triangulation into a geometricone. A face of a combinatorial pseudo-triangulationis called degenerate if it contains edges which appeartwice on the boundary of this face.Proposition 3 [6, Corollary 2] The following prop-erties are equivalent for a combinatorial pseudo-triangulation G: (1) G can be stretched to becomea pseudo-triangulation. (2) G has the generalizedLaman property. (3) G has no degenerate faces andevery subgraph of G with at least three vertices has atleast three corners of rst type.Since, by denition, combinatorial 4-PPTs have nodegenerate faces, we can use Proposition 3 to concludethe following.Theorem 4 Every combinatorial 4-PPT can bestretched to become a 4-PPT with the given assign-ment of angles. Furthermore, combinatorial 4-PPTshave the generalized Laman property.Note that there exist non-stretchable combinato-rial pointed pseudo-triangulations with faces of sizeat most 5. See Figure 1. There and in the forthcom-ing gures, arcs denote angles labeled as reex.Figure 1: A non-stretchable combinatorial pointedpseudo-triangulation [6].3 FlipsIn the following we focus on combinatorial 4-PPTswith a xed triangular outer face. For such a com-binatorial 4-PPT, Corollary 2 implies that there isonly one interior triangular face. Before dening ipsbetween combinatorial 4-PPTs, we make some obser-vations about their geometric counterpart.Geometric 4-PPTs with triangular convex hull alsohave only one interior triangle. Furthermore, everyedge of the triangle (except for those being part of theconvex hull) is ippable [7]. Observe that the removalof the edge e to be ipped merges the triangle andthe 4-face adjacent at e into a 5-face, which mightbe degenerate if the triangle and the 4-face share twoedges. See Figure 2. Note that this is the only casein which the triangle and the 4-face can share threevertices, as there are no multiple edges in geometricgraphs.132XV Spanish Meeting on Computational Geometry, June 26-28, 2013Figure 2: Geometric ip of an edge of a triangle. Inthe lower case, removal of the ipped edge gives adegenerate 5-face.Similar to the geometric case, we consider ips ofan edge e of the (unique) interior triangular face T ina combinatorial 4-PPT (with triangular outer face):Consider the 4-face F sharing e with T . A ip of econsists in replacing e by another edge e′ such that(1) e′ splits (T∪F )\e into a triangular face T ′ and a4-face F ′ and (2) the result is a combinatorial 4-PPT.In particular, and in contrast to the geometric case, inthe combinatorial setting we have to explicitly avoidmultiple edges and thus to ensure that the edge e′we insert is not already contained in the combinato-rial 4-PPT (as an edge outside T∪F ). The followinglemma shows that every interior edge of the interiortriangular face can be ipped.Lemma 5 In a combinatorial 4-PPT, every edge eof an interior triangular face that is not an edge ofthe outer face is ippable. Furthermore: (1) If theremoval of e results in a degenerate 5-face, then thereis a unique valid ip for e. (2) If removing e resultsin a non-degenerate 5-face, then there are at least twovalid ips for e.Observe that, given a combinatorial ip betweentwo combinatorial 4-PPTs, by Theorem 4 we knowthat both of them can be stretched into geometric 4-PPTs with straight edges. However, it might not bepossible to use the same geometric embedding for thevertices in both of them.4 Flip graph connectivityLemma 6 For a given combinatorial 4-PPT with tri-angular outer face and for any edge b of this outerface, there is a sequence of ips resulting in a combi-natorial 4-PPT whose interior triangular face is inci-dent to b.Once the interior triangular face is incident to anedge b of the outer face, the next step will be ippingaway interior edges incident to one endpoint of b.Lemma 7 Given a combinatorial 4-PPT with tri-angular outer face, in which the interior triangularface T is incident to the edge b of the outer face, thereis a sequence of ips resulting in a combinatorial 4-PPT in which the endpoint v of b = uv has no interiorincident edges.Proof. We describe a ip sequence that ips all in-ner edges incident to v. This ip sequence can bepartitioned into two phases and some cases. Let thevertices neighbored to the vertex v be ordered radi-ally around v, starting with u. In each case, let thevertices in that order be u = w0, . . . , wk.Phase 1: During this phase, the inner triangular faceT has uv as a side, i.e., T = vuw1. We distinguishthree dierent cases:Case 1: vw1 is the only inner edge incidentto v, i.e., k = 2. If T is incident to only one 4-faceF (i.e., T∪F is degenerate), we can ip the edge vw1and are done. Otherwise, let the 4-face F incident tovw1 be vw1sw2. See Figure 3. The reex angle insideF is either at s or w1. If it is at s, we ip vw1 to w0s,obtaining the 4-face vw0sw2. Otherwise, the reexangle is at w1 and we ip vw1 to w1w2, obtaining the4-face vw0w1w2. Either way, the degree of v is 2 andwe are done.vw1svw1w2 w0u = w0vw1vsu = w0 w2w2w2w0sw1sFigure 3: Phase 1, Case 1: Only one interior edge isincident to v.Case 2: at least two inner edges are incidentto v and there does not exist an edge w0w2. SeeFigure 4. Since the reex angle of v is at the outer facewe can replace the edge vw1 by w0w2. This reducesthe degree of v by one. The inner triangular face isagain adjacent to w0v, and we remain in Phase 1.w1w2v vu = w0 w0w2w1Figure 4: Phase 1, Case 2: Several interior edges areincident to v and w0w2 does not exist.133Flips in combinatorial pointed pseudo-triangulations with face degree at most four (extended abstract)Case 3: at least two inner edges are incidentto v and there exists an edge w0w2. See Figure 5.If the two inner edges of T are incident to a single4-face, we have a degenerate case; we ip the edgew0w1 to w1w2, making vw1w2 the inner triangularface. Otherwise, let the 4-face F incident to vw1 bevw1sw2; we ip vw1 to vs (this is possible since ifvs already existed, it would have to cross the cycleuw2sw1). Either way, the ip does not reduce thedegree of v, but the inner triangular face is now insidethe 3-cycle vw0w2. We switch to Phase 2.w1w2 w2w2w1w2s su = w0u = w0w0w0v vv vw1w1Figure 5: Phase 1, Case 3: The possible transitionsto Phase 2.Phase 2: During this phase, the inner triangular faceis vw1w2, and w1 stays xed for the whole phase.Further, we know that w1 was enclosed by a 3-cycle(at the transition to this phase), which implies thatthere are no edges from w1 to wi for any i ≥ 2. Wedecrease the degree of v in the following manner.Case 1: there is a 4-face F incident to vw2.There cannot be an edge w1w3 since w1 was enclosedby a 3-cycle. Further, the reex angle of F is notat v. Hence, we can ip vw2 to w1w3, which reducesthe degree of v and we remain in Phase 2, with vw1w3being the new inner triangular face.Case 2: there is no 4-face incident to vw2, i.e.,k = 2. This case is symmetric to Case 1 of Phase 1.The edge vw1 is ipped in one of the two describedways, reducing the degree of v to 2 and thus endingthe process. Theorem 8 The graph of ips in combinatorial 4-PPTs with n vertices and triangular outer face is con-nected and has diameter O(n2).Proof. Given such a combinatorial 4-PPT, follow thesteps in Lemmas 6 and 7, then use induction for thecombinatorial 4-PPT obtained by removing v. Thisleads to the unique canonical combinatorial 4-PPTwith triangular outer face, where two of the verticesin the outer face are adjacent to all other vertices,while the third one has degree 2. See Figure 6.Figure 6: A canonical combinatorial 4-PPT.Furthermore, the number of ips needed in Lemmas 6and 7 is at most linear in the number of vertices ofthe combinatorial 4-PPT. The presented basic case of combinatorial 4-PPTswith triangular outer face is extendible to an arbi-trarily sized outer face, to labeled vertices, and alsoto the general case of combinatorial 4-PPTs with anarbitrarily sized outer face on labeled vertices. Elabo-rating on these extensions would go beyond the scopeof this extended abstract, though. Details (and omit-ted proofs) can be found in a forthcoming full version.Acknowledgments. This work was initiated duringthe 9th European Research Week on Geometric Graphsand Pseudo-Triangulations, held May 1418, 2012 in Al-calá de Henares, Spain. We thank Vincent Pilaud, PedroRamos, and André Schulz for helpful comments.References[1] O. Aichholzer, F. Aurenhammer, T. Hackl,C. Huemer, A. Pilz, and B. Vogtenhuber. 3-colorability of pseudo-triangulations. In EuroCG2010, pages 2124, 2010.[2] O. Aichholzer, F. Aurenhammer, H. Krasser, andP. Braÿ. Pseudotriangulations from surfaces anda novel type of edge ip. SIAM J. Comput.,32(6):16211653, 2003.[3] S. Bereg. Transforming pseudo-triangulations.Inf. Process. Lett., 90(3):141145, 2004.[4] P. Bose and F. Hurtado. Flips in planar graphs.Comput. Geom., 42(1):6080, 2009.[5] L. Kettner, D. Kirkpatrick, A. Mantler,J. Snoeyink, B. Speckmann, and F. Takeuchi.Tight degree bounds for pseudo-triangulations ofpoints. Comput. Geom., 25(12):312, 2003.[6] D. Orden, F. Santos, B. Servatius, and H. Ser-vatius. Combinatorial pseudo-triangulations. Dis-crete Math., 307(35):554566, 2007.[7] G. Rote, F. Santos, and I. Streinu. Pseudo-triangulations  a survey. Contemp. Math. AMS,2008.134

Flips in combinatorial pointed pseudo-triangulations with face degree at most four

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Flips in combinatorial pointed pseudo-triangulations with face degree at most four

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