We exhibit a family of graphs which can be realized as pseudo-visibility graphs of pseudo-polygons, but not of straight-line polygons. The example is based on the characterization of vertex-edge pseudo-visibility graphs of O\u27Rourke and Streinu [Proc. ACM Symp. Comput. Geometry, Nice, France, 1997, pp. 119-128] and extends a recent result of the author [Proc. ACM Symp. Comput. Geometry, Miami Beach, 1999, pp. 274-280] on non-stretchable vertex-edge visibility graphs. We construct a pseudo-visibility graph for which there exists a unique compatible vertex-edge visibility graph, which is then shown to be non-stretchable. The construction is then extended to an infinite family. © 2004 Elsevier B.V

Streinu, Ileana

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Smith ScholarWorks Computer Science: Faculty Publications Computer Science 6-1-2005 Non-Stretchable Pseudo-Visibility Graphs Ileana Streinu Smith College, istreinu@smith.edu Follow this and additional works at: https://scholarworks.smith.edu/csc_facpubs  Part of the Computer Sciences Commons Recommended Citation Streinu, Ileana, "Non-Stretchable Pseudo-Visibility Graphs" (2005). Computer Science: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/csc_facpubs/309 This Article has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact scholarworks@smith.edu Non-Stretchable Pseudo-Visibility GraphsIleana StreinuDept. of Computer Science,Smith College,Northampton, MA 01063, USA.streinu@cs.smith.edu. AbstractWe exhibit a family of graphs which can be realized aspseudo-visibility graphs of pseudo-polygons, but notof straight-line polygons. The construction is based onthe characterization of vertex-edge pseudo-visibilitygraphs of O'Rourke and Streinu[ORS96] and extendsrecent results on non-stretchable vertex-edge visibil-ity graphs of Streinu [Str99]. We show that there isa pseudo-visibility graphs for which there exists onlyone of vertex-edge visibility graph compatible with it,which is then shown to be non-stretchable. The con-struction is then extended to an innite family.1 IntroductionCharacterizing visibility graphs is a problem with adistinguished history (Ghosh[Gho88], Everett[Ev90],Abello and Kumar[AK95]), but so far several attemptsto give good sets of conditions have been proved in-sucient.A dierent approach, introduced by O'Rourke andStreinu [ORS96] is to separate the combinatorial as-pects of the problem from the questions of stretchabil-ity (known to be notoriously hard for pseudo-line ar-rangements, cf. Mnev [Mn91] and Shor [Sh91]). Theyhave introduced two new concepts: vertex-edge vis-ibility graphs [ORS98] and pseudo-visibility [ORS96],and gave a complete combinatorial characterizationof vertex-edge pseudo-visibility graphs. However, itis not clear a priori that the new class is any largerthan just the class of straight-line vertex-edge visibil-ity graphs, since it is conceivable that all such graphscan be realized with straight line edges (just as theResearch partially supported by NSF RUI grant CCR-9731804 and a Smith College Picker Fellowship for FacultyDevelopment.class of planar graphs is realizable in this manner).In [Str99] a whole class of non-stretchable vertex-edgepseudo-visibility graphs is exhibited, thus settling thisquestion. Moreover, it is shown that the stretchabilityquestion can in fact be decided eciently for a classof vertex-edge pseudo-visibility graphs which includesthese examples.The original question was for visibility graphs, notvertex-edge visibility graphs. Since vertex-edge pseudo-visibility graphs contain more information than pseudo-visibility graphs, it is possible to have several vertex-edge pseudo-visibility graphs compatible with a givenpseudo-visibility graph: some may be stretchable, somenot. We are interested in the question: are there anypseudo-visibility graphs for which none of the com-patible vertex-edge pseudo-visibility graph is stretch-able? The visibility graphs of the non-stretchablepseudo-polygons from [Str99] are in fact compatiblewith straight line polygons, so the same family of ex-amples does not work directly.In this paper we exhibit a slightly more involvedexample of a pseudo-polygon with the property thatits pseudo-visibility graph uniquely induces a vertex-edge visibility graph, which is then shown to be non-stretchable. This provides a strong separation be-tween straight-line and pseudo visibility graphs. Theexample is then extended to an innite family.2 PreliminariesAbbreviations: Wemay abbreviate the prex pseudoby p- (as in p-line for pseudo-line), vertex-edge pseudo-visibility graph by ve-graph, pseudo-visibility graph byv-graph and generalized conguration of points by gcp.We use ccw for counter-clockwise.An arrangement of pseudolines L is a collectionof simple curves, each of which separates the plane,such that each pair of p-lines of L meet in exactly onepoint, where they cross.Denition 2.1 Let V = fv0; v1; : : : ; vn 1g be a setof points in the Euclidean plane IR2, and let L be anarrangement of n2pseudolines such that every pair ofpoints vi and vj lie on exactly one pseudoline lij 2 L,and each pseudoline in L contains exactly two points ofV . Then the pair (V;L) is a generalized conguationof points in general position.Two points a and b on a pseudoline l 2 L determinea unique (closed) segment ab consisting of those pointson l that lie between the two points. For 0  i  n 1,let ei = vivi+1 be the segment determined by vi andvi+1 on li;i+1 .1Denition 2.2 The segments ei = vivi+1 form a pseudo-polygon i:1. The intersection of each pair of segments adja-cent in the cyclic ordering is the single pointshared between them: ei \ ei+1 = vi+1, for alli = 0; 1; : : : ; n  1.2. Nonadjacent segments do not intersect: ei\ej =;, for all j 6= i+ 1.A p-polygon is a simple closed Jordan curve andseparates the plane into two regions. We assume with-out loss of generality that the vertices of the p-polygonare numbered in ccw order, i.e. that the interior of thepolygon lies to the left as the boundary is traversed inthis order.Pseudo-visibility is determined by the underlyingarrangement L: lines-of-sight are along pseudolines inL.Denition 2.3 Vertex vi sees vertex vj (vi $ vj) ieither vi = vj, or they lie on a line lij 2 L and thesegment vivj is nowhere exterior to P .12345677910Figure 1: A non-stretchable generalized congurationof points.Denition 2.4 The vertex-vertex pseudo-visibility graph(v-graph)GV (P ) of a p-polygon is a labeled graph withnode set V , and an arc between two vertices i theycan see one another (according to Def. 2.3).1All index arithmetic is mod n throughout the paper.We will often abbreviate GV (P ) to GV . Note that GVis Hamiltonian: the arcs corresponding to the polygonboundary form a Hamiltonian circuit (v0; : : : ; vn 1).And also note that since GV is labeled by V , whichwe assumed was labeled in a ccw boundary traver-sal order, the Hamiltonian circuit is provided by thelabeling of the graph.To dene vertex-edge pseudo-visibility we need todene when a vertex sees an edge. This is based on thenotion of a \witness" for a visible pair. Let rij  lijbe the ray directed from vj not including vi, closed atvj.Denition 2.5 Vertex vj is a witness for the vertex-edge pair (vi; e) (and we say that vi sees edge e) ieither1. vi is an endpoint of e, and vj is also (here wepermit vj = vi); or2. vi is not an endpoint of e, and(a) vi sees vj; and(b) the ray rij intersects e at a point p,(c) either vj = p, or the segment vjp is nowhereexterior.We will refer to the line lij in the above denition asthe witness line.12345677910Figure 2: A non-stretchable pseudo-polygon.Denition 2.6 The vertex-edge pseudo-visibility graphGV E of a polygon is a labeled bipartite graph with nodenode set V [E, and an arc between v 2 V and e 2 Ei v can see e (according to Def. 2.5).Notation: Let P (i; j) be the open boundary inter-val containing all verices and edges of P encounteredin a ccw traversal of the boundary of P from vi tovj. Similarly we dene P [i; j), P (i; j] and P [i; j] toinclude one or both endpoints of the interval.The following lemma has been proved in [ORS96]:Lemma 2.7 If vk sees non-adjacent edges ei and ejand no edge between, vk 2 P [j+1; i], then exactly oneof Case A or B holds:A 1. vk sees vi+1 but not vj.2. vi+1 is the right-witness for (vk; ej).3. vi+1 sees ej but vj does not see ei.B 1. vk sees vj but not vi+1.2. vj is the left-witness for (vk; ei).3. vj sees ei but vi+1 does not see ejOne more concept needed is that of a \pocket":Denition 2.8 If vi sees ej and vr and vl are theright and left witnesses respectively, then P [i; r) andP (l; i] are the right and left near pockets, and P (r; j]and P [j + 1; l) are the right and left far pockets ofvi ! ej respectively.The following lemma has been proved in [ORS96].Lemma 2.9 If vi sees ej and vr and vl are the rightand left witnesses respectively, then1. No vertex in the right near pocket sees an edgein the right far pocket.2. No vertex in the right far pocket sees an edge inthe right near pocket.Symmetric claims hold for the left pockets.Lemma 2.10 If vi sees ej and vr and vl are the rightand left witnesses respectively, then vr is an articula-tion point of the subgraph of GVE induced by P [i; j],and symmetrically vl is an articulation point of thesubgraph induced by P [j+ 1; i].Theorem 2.11 If GV E is the vertex-edge visibilitygraph of a pseudo-polygon P , then it satises thesetwo properties:1. If vk sees non-adjacent edges ei and ej and noedge between, vk 2 P [j + 1; i], then exactly oneof these holds:A. (vi+1; ej) 2 GV E , orB. (vj; ei) 2 GVE .2. In the two cases above, additionally:A. vi+1 is an articulation point of the subgraphof GVE induced by P [k; j].B. vj is an articulation point of the subgraph ofGVE induced by P [j + 1; k].It has been shown in [ORS96] that these propertiesprovide a complete characterization of vertex-edge pseudo-visibility graphs.We now turn to stretchability questions. The basisof our construction comes from a classical exampleof a non-realizable allowable sequence, the so-callednon-realizable pentagon (see [GP93]). The ve centralpoints in Figure 1 are pairwise connected by pseudo-lines. We force these pseudo-lines to cross such thatwe can then place the other exterior ve points. Wethen draw extra pseudo-lines connecting all these pairsof points. This can be achieved in several ways, buteach is a non-stretchable conguration of points.To get an exampel of a non-stretchable pseudo-polygon, as in [Str99], we place a pseudo-polygon ontop of this generalized conguration of points, as inFigure 2. No matter how the other pseudo-lines inthe conguration meet, the basic internal structureremains the same: they all have the same vertex-edgevisibility graph. Also, any pseudo-polygon with thisve-graph has to contain the non-realizable pentagonas un underlying subarrangement of its congurationof points.3 An unstretchable vertex-vertex pseudo-visibilitygraphOur goal is to extend this example to a non-realizablepseudo v-graph. We would like to get a v-graph withonly one compatible ve-graph - the realizable one. Butunfortunately the v-graph underlying the unstretch-able ve-graph (Figure ??) has both realizable andunrealizable compatible ve-graphs.So we'll have to work harder to get our example.Notice that in general a v-graph can have many com-patible ve-graphs (exponentially many, as shown inthe full paper).Figure 3: The gadget used in constructing the non-realizable v-graph.The idea for obtaining a non-stretchable v-graphis to brake the symmetry, forcing the v-graph to haveonly one ve-graph. This ve-graph should lead to thenon-stretchable pentagon. The construction is morecomplicated. It is based on the gadget in Fig. 3. Thegadget is repeated at each of the ve exterior verticesof the pentagon. The shick solid edges represent thecommon part of the v-graph, corresponding to the cen-tral mutually visible ve vertices of the non-realizablepentagon. The thin solid and the thin dashed edgeson the top are the actual gadget. The dashed edgesare the symmetry brakers.Let's prove that there is only one compatible ve-graph for this v-graph.The argument is based on the properties of ve-graphs listed in the Preliminaries. Let's denote the rel-evant vertices of the gadget as in Figure 4: v1; v2; b2; w2; t; w1and b1.tw1w2b 2b1v1 v2Figure 4: The labeled gadget.Then since v1 sees w2, v1 sees b1 and no other ver-tex between these tow (in the ccw order induced bythe boundary of the polygon), it follows that eitherb1 or w2 is an articulation point for t, when visibilityfrom v1 is considered. But w2 can't be, since b2 sees t(otherwise we would get a contradiction of the artic-ulation point property of ve-graphs for b2 in the nearpocket, t in the far pocket, with articulation point w2and visibility from v1.So it follows that b1 has to be articulation pointfor t with visibility from v1. But then it follows thatthe pseudo-line through v1b1 extends to intersect theedge tw2 (as in Figure 4).A similar argument holds for the pseudo-line throughv2b2. When we repeat this for all the ve gadgets, werealize that we forced the unrealizable pentagon fromFigure 1 as part of the underlying conguration ofpoints for the ve-graph!This completes the un-stretchable v-graph exam-ple. In the full paper we also show that this examplecan be extended to an innite family.4 ConclusionWe have shown that the class of pseudo v-graphs isstrictly larger than the class of straight-line v-graphs.This result, together with the main characterizationfrom [ORS96], yields a number of open problems forfurther research, which are described in the full paper.References[AK95] J. Abello and K. Kumar. Visibility graphsand oriented matroids. In R. Tamassiaand I. G. Tollis, editors, Graph Drawing(Proc. GD '94), volume 894 of LectureNotes in Computer Science, pages 147{158. Springer-Verlag, 1995.[Ev90] H. Everett. Visibility Graph Recognition.Ph.D. thesis, Computer Science Depart-ment, University of Toronto, 1989.[Gho88] S. K. Ghosh. On recognizing and char-acterizing visibility graphs of simple poly-gons. In Proc. 1st Scand. Workshop Al-gorithm Theory, volume 318 of LectureNotes in Computer Science, pages 96{104.Springer-Verlag, 1988.[GP80] J.E. Goodman and R. Pollack. Onthe combinatorial classication of non-degenerate congurations in the plane. J.Combinat. Theory, ser. A 29 (1980), 220-235.[GP93] J.E. Goodman and R. Pollack. Allowablesequences and Order Types in Discrete andComputational Geometry in New Trendsin Discrete and Computational Geometry,J. Pach (ed.), Springer Verlag, 1993[Mn91] N. Mnev. The Universality Theorem onthe Oriented Matroid Stratication of theSpace of Real Matrices. in Discrete andComputational geometry, DIMACS Spe-cial Year (eds. J. Goodman, R. Polalck andW. Steiger), AMS/ACM DIMACS Seriesin Discrete Math. and Theoretical Com-puter Science, vol. 6, 1991.[ORS96] J. O'Rourke, I. Streinu. Vertex-EdgePseudo-Visibility Graphs: Characteriza-tion and Recognition, Proc. ACM Symp.Comput. Geometry, Nice (France), 1997.[ORS98] J. O'Rourke, I. Streinu. The Vertex-EdgeVisibility Graph of a Polygon, Comput.Geom. Theory Appl. 10 (105-120), 1998.[Sh91] P. Shor. Stretchability of Pseudolinesis NP-Hard. in Applied Geometry andDiscrete Mathematics, the Victor KleeFestscrift, P. Gritzmann, B. Sturmfels(eds.), DIMACS Series in Discrete Math.and Theoretical Computer Science, vol. 4,1991.[Str99] I. Streinu. Stretchability of Star-likePseudo-Polygons, Proc. ACM Symp. Com-put. Geometry, Miami Beach, 1999 (to ap-pear).

Non-Stretchable Pseudo-Visibility Graphs

AbstractWe exhibit a family of graphs which can be realized as pseudo-visibility graphs of pseudo-polygons, but not of straight-line polygons. The example is based on the characterization of vertex-edge pseudo-visibility graphs of O'Rourke and Streinu [Proc. ACM Symp. Comput. Geometry, Nice, France, 1997, pp. 119–128] and extends a recent result of the author [Proc. ACM Symp. Comput. Geometry, Miami Beach, 1999, pp. 274–280] on non-stretchable vertex-edge visibility graphs. We construct a pseudo-visibility graph for which there exists a unique compatible vertex-edge visibility graph, which is then shown to be non-stretchable. The construction is then extended to an infinite family

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Non-stretchable pseudo-visibility graphs 

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Non-Stretchable Pseudo-Visibility Graphs

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