It is well known that, given n red points and n blue points on a circle, it is not always possible to find a plane geometric Hamiltonian alternating path. In this work we prove that if we relax the constraint on the path from being plane to being 1-plane, then the problem always has a solution, and even a Hamiltonian alternating cycle can be obtained on all instances. We also extend this kind of result to other configurations and provide remarks on similar problems.Ministerio de Economía y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e InnovaciónJunta de Andalucía (Consejería de Innovación, Ciencia y Empresa

Claverol Aguas, Mercè

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia

Garijo Royo, Delia (Coordinador)

Hurtado Díaz, Ferran

Lara Cuevas, María Dolores

Márquez Pérez, Alberto (Coordinador)

Seara Ojea, Carlos

Urrutia Galicia, Jorge (Coordinador)

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013The alternating path problem revisitedMercè Claverol1, Delia Garijo2, Ferran Hurtado1, Dolores Lara3, and Carlos Seara11,3Universitat Politècnica de Catalunya, Barcelona, Spain2Universidad de Sevilla, SpainAbstractIt is well known that, given n red points and n bluepoints on a circle, it is not always possible to nd aplane geometric Hamiltonian alternating path. In thiswork we prove that if we relax the constraint on thepath from being plane to being 1-plane, then the prob-lem always has a solution, and even a Hamiltonian al-ternating cycle can be obtained on all instances. Wealso extend this kind of result to other congurationsand provide remarks on similar problems.IntroductionA geometric graph is a graph drawn in the planewhose vertex set is a set of points and whose edgesare straight-line segments connecting pairs of vertices.Two edges of a geometric graph cross if they havean intersection point lying in the relative interior ofboth edges. A plane geometric graph is a geometricgraph without any edge crossings. A 1-plane geomet-ric graph is a geometric graph in which every edgehas at most one crossing. Notice that the terms planegraph and 1-plane graph refer to a geometric object,while to be planar or 1-planar are properties of theunderlying abstract graph. We use here standard no-tation for geometric graphs as in [3] and [12].Let P be a set of points in the plane in generalposition (i.e., no three points are collinear), and letCH(P ) denote its convex hull. A geometric spanningtree on P , generically denoted by tree(P ), is any span-ning tree on P whose edges are straight-line segmentsconnecting two points on P . Observe that if |P | = 1,the tree on P is a single vertex. When the tree isa path, we write path(P ). The geometric completegraph K(P ) on P is the complete geometric graphwith vertex set P . Notice that tree(P ) is a spanningtree of K(P ).1Email: merce@ma4.upc.edu, ferran.hurtado@upc.edu,carlos.seara@upc.edu. Partially supported by projectsMINECO MTM2012-30951, Gen. Cat. DGR2009SGR1040,and ESF EUROCORES programme EuroGIGA, CRP Com-PoSe: MICINN Project EUI-EURC-2011-4306.2Email: dgarijo@us.es. Partially supported by projects2009/FQM-164 and 2010/FQM-164.3Email: maria.dolores.lara@upc.edu.Let R and B be two disjoint sets of red and bluepoints in the plane such that no three points of R∪Blie on the same line. The geometric complete bipartitegraph K(R,B) is the graph with vertex set R∪B andwhose edges are all the straight-line segments connect-ing any point in R and any point in B. A line segmentdened by two red points is a red segment, and one de-ned by two blue points is a blue segment. More gen-erally, an edge is said to be monochromatic when thetwo endpoints have the same color, and bichromaticotherwise. The intersections between red segmentsand blue segments are called bichromatic crossings,and those between segments having the same colorare called monochromatic crossings.Problems on adding edges to a given graph to ob-tain a new graph with some desirable properties are,in general, called augmentation problems. Amongthese, plane augmentation considers an initial planegraph G = (V,E) (possibly empty, i.e., only the pointset V is given) that has to be augmented to anotherplane supergraph G′ = (V,E ∪E′) by adding a set E′of edges to G, see the survey [7].In this work we focus on problems in which the ini-tial point set is a bicolored set R ∪ B. This familyof problems has attracted a substantial amount of re-search, see for instance the surveys [8, 7].We rst consider alternating graphs, i.e., those inwhich every edge is bichromatic. A well known fact[2, 1, 10, 11] is that given n red points and n bluepoints on a circle (equivalently, in convex position),one cannot always obtain a plane geometric Hamilto-nian alternating path. In this paper we prove that, ifwe relax the constrain on the geometric Hamiltonianalternating path from being plane to being 1-plane,then a solution always exists, even yielding strongerproperties. We also show that the same result holdsfor some other congurations. These results appearin Section 1.Regarding monochromatic graphs, i.e, graphs inwhich every edge is monochromatic, it is easy tosee that one cannot always construct a plane perfectmatching in K(R)∪K(B). A trivial example is givenby the vertices of a convex quadrilateral in which twoopposite vertices are colored red and the other two arecolored blue. The same example shows that it is notalways possible to obtain two geometric monochro-115The alternating path problem revisitedmatic spanning trees, tree(R) and tree(B), such thattheir union is plane.The proven nonexistence of plane congurationshad already suggested to researchers to allow somerelaxation in the constraint, but the focus was puton constructing geometric graphs having globally fewcrossings [9, 13]. However, one of the constructions in[13] is in fact a 1-plane graph. We include some re-marks on that particular result and its consequencesin Section 2.We conclude in Section 3 with some additional com-ments.1 Alternating graphs: paths andcyclesIn this section we study geometric alternating span-ning graphs on R ∪ B, i.e., spanning subgraphs ofK(R,B). We focus on geometric Hamiltonian alter-nating paths and cycles, which visit red points andblue points alternately. Notice that when there are nocrossings at all, geometric Hamiltonian paths and cy-cles are also called simple polygonals and simple poly-gons, respectively.1.1 Convex positionThe problem of restricting the bicolored point set tolay on a circle has attracted a lot of attention. It iseasy to see that there are sets R and B with the samenumber of points, say n, such that R∪B is in convexposition, and a plane geometric Hamiltonian alter-nating path on R ∪ B cannot exist. Erd®s (see [10])proposed in 1989 to study the value ℓ(n) such thatno matter how the colors are distributed a plane geo-metric alternating path of length at least ℓ(n) alwaysexists. About the same time, Akiyama and Urrutia [2]considered independently the same problem: theyproved a necessary and sucient condition for the ex-istence of a plane geometric Hamiltonian alternatingpath and derived an O(n2) time algorithm to nd one,if it exists. Abellanas et al. [1], and independentlyKyn£l et al. [10], proved that ℓ(n) ≤ 43n + O(√n),and Cibulka et al. [4] showed that ℓ(n) ≥ n+Ω(√n).These bounds are the best to date. Remind that thetotal number of points is 2n. Also, notice that weslightly abuse the notation by using inequalities, andthat the two bounds have to be read together, notindependently. For more information and details, seeMészáros' PhD Thesis [11].We next show that if R ∪ B is in convex positionthen a 1-plane geometric Hamiltonian alternating cy-cle not only a path can always be drawn. The fol-lowing lemma is the key tool.Lemma 1 Let R and B be two disjoint sets of redand blue points in the plane such that R ∪ B is inconvex position, and |R| = |B| = n ≥ 2. Let S be aset of disjoint bichromatic segments on the boundaryof the convex hull of R ∪ B, and |S| = s ≥ 2. Then,there exists a 1-plane geometric Hamiltonian alternat-ing cycle on R ∪ B that contains each segment of Sas an edge.If R ∪ B is in convex position and |R| = |B| =n ≥ 2 then there are at least two disjoint bichromaticsegments on the boundary of CH(R ∪ B), say s1, s2.Let S = {s1, s2}. By Lemma 1, one can draw a 1-plane geometric Hamiltonian alternating cycle on R∪B that contains each segment of S as an edge. Thisargument proves our main result in this section.Theorem 2 Let R and B be two disjoint sets of redand blue points in the plane such that R∪B is in con-vex position, and |R| = |B| = n ≥ 2. Then, there ex-ists a 1-plane geometric Hamiltonian alternating cycleon R ∪B.1.2 Double chainWe next consider the problem of drawing a 1-planegeometric Hamiltonian alternating cycle on a doublechain whose points are colored red and blue.The double chain (formally dened below) isa conguration that has been intensively stud-ied since it admits many triangulations, manypolygonizations, many crossing-free matchings, etc.,and for several families of graphs yields themaximum number known to date of such pos-sible congurations among point sets with thesame cardinality (see http://www.cs.tau.ac.il/ shef-fera/counting/PlaneGraphs.html).A double chain (C1, C2) consists of two oppositeconvex chains C1 and C2, facing each other, such thatthe convex hull of C1∪C2 is a quadrilateral, each pointof C2 lies strictly below every line determined by twopoints of C1, and each point of C1 lies strictly aboveevery line determined by two points of C2. When thepoints of (C1, C2) are colored red and blue, (C1, C2)is said to be a bicolored double chain (see Figure 1),and ri, bi denote the number of red and blue points,respectively, of Ci for i = 1, 2. Note that the sizes ofC1 and C2 may be dierent.1C2CFigure 1: A bicolored double chain (C1, C2).116XV Spanish Meeting on Computational Geometry, June 26-28, 2013Cibulka et al. [4] proved the following theorem:Theorem 3 [4] (i) If |Ci| ≥ 15 (|C1| + |C2|) for i =1, 2, then there exists a non-crossing geometric Hamil-tonian alternating path on (C1, C2); (ii) there exist bi-colored double chains in which one of the chains con-tains at most 1/29 of all the points, which do not ad-mit a non-crossing geometric Hamiltonian alternatingpath.Theorem 4 below states that when r1+r2 = b1+b2,allowing at most one crossing per edge is strongenough to let us always draw a geometric Hamilto-nian alternating cycle on (C1, C2). The main idea toobtain this cycle is to use Theorem 2 to constructtwo 1-plane geometric alternating cycles Λ1 and Λ2in CH(C1) and CH(C2), respectively, and to connectthem drawing another 1-plane geometric alternatingcycle Λ in the exterior of CH(C1) and CH(C2). Thisprocess is described next.Let Ei, i = 1, 2, be the set of edges in CH(Ci) thatconnect consecutive points of Ci. Suppose rst that3 ≤ b1 + 1 < r1 and 3 ≤ r2 + 1 < b2. Then thereexist at least two monochromatic edges in E1 ∪E2: ared edge rr′ ∈ E1 and a blue edge bb′ ∈ E2. Contractthem obtaining a red point r′′ and a blue point b′′.By Theorem 2, we can draw a cycle Λ1 on the pointset formed by the b1 blue points of C1, the red pointr′′, and b1 − 1 red points of C1 \ {r, r′}. Analogously,Λ2 is constructed on the point set formed by the r2red points of C2, the blue point b′′, and r2 − 1 bluepoints of C2 \ {b, b′}. Finally, the cycle Λ is drawn, asin Figure 2, on the remaining r1 − b1 − 1 red pointsof C1, the remaining b2 − r2 − 1 = r1 − b1 − 1 bluepoints of C2, and the points r′′ and b′′. Observe thatr′′ and b′′ connect Λ with Λ1 and Λ2, respectively.By reversing the contraction and deleting the edgesrr′ and bb′, we "open" the three cycles obtaining thedesired cycle on (C1, C2).Note that the preceding argument can easily beadapted for b1 or r2 equal to zero or one. Observealso that if all the edges in E1 (analogous for E2) arebichromatic then either r1 = b1 (and so r2 = b2) orr1 = b1 + 1 (and b2 = r2 + 1). For these values, evenif there are monochromatic edges in E1, we need touse slightly dierent arguments which are omitted forthe sake of brevity.Theorem 4 Let R and B be the sets of red andblue points of a bicolored double chain (C1, C2), and|R| = |B| ≥ 2. Then, there exists a 1-plane geometricHamiltonian alternating cycle on (C1, C2).1.3 General positionThe positive results for convex position and for thedouble chain make one wonder whether a similar re-sult holds for any set of points in general position:Question 1 Let R and B be any two disjoint setsof red and blue points in the plane such that no threepoints of R∪B lie on the same line, and |B| = |R| ≥ 2.Does there always exist a 1-plane geometric Hamilto-nian alternating cycle on R ∪B?We do not know the answer to the preceding ques-tion. Geometric Hamiltonian cycles with few cross-ings were obtained by Kaneko et al. [9], who gavea tight upper bound of |R| − 1 for the number ofcrossings of a geometric Hamiltonian alternating cy-cle. Figure 2 illustrates a conguration R ∪ B forwhich this upper bound is best possible.Figure 2: A 1-plane geometric Hamiltonian alternat-ing cycle, on a point set R∪B, with |R|−1 crossings.To be precise, Kaneko et al. [9] proved the followingresult.Theorem 5 [9] Let R and B be two disjoint sets ofpoints in the plane such that |R| = |B| and no threepoints of R ∪ B are on the same line. Then we candraw a geometric Hamiltonian alternating cycle onR ∪ B which has at most |R| − 1 crossings. More-over there exist congurations R ∪ B for which thisupper bound |R| − 1 is the best possible.To prove this theorem, the authors use several lem-mas that we have carefully examined to see whetherit is possible to adapt their proof to obtain a 1-planegraph. However, as far as we can see, there are casesin which the connection of the paths that exist byinduction is not necessarily 1-plane.It is unclear to us which is the right answer to Ques-tion 1, yet our study leads us to believe that it isnegative in general, yet positive if only a Hamiltonianpath is required, which we state as a conjecture:Conjecture 1 Let R and B be two disjoint sets of redand blue points in the plane such that no three pointsof R∪B lie on the same line, and |B| ≤ |R| ≤ |B|+1.There always exists a 1-plane geometric Hamiltonianalternating path on R ∪B.Let us mention that this is ongoing research, cur-rently focusing on the preceding conjecture, which wehope to answer in the next version of this paper.117The alternating path problem revisited2 A remark on monochromaticgraphsIn this section we would like to remark that a resultobtained by Tokunaga in 1996 can be rephrased interms of 1-plane graphs, hence giving support to theidea that exploring this relaxation may be worth theeort for more problems.On one hand, it is not always possible to constructa non-crossing perfect matching in K(R) ∪ K(B) asproved by Dumitrescu and Steiger [6]. The originalresult was improved by Dumitrescu and Kaye [5], whoproved that for given R and B, with |R| + |B| = n,there always exists a non-crossing matching inK(R)∪K(B) which covers at least 0.8571 · n points of R ∪B, while for some congurations every non-crossingmatching in K(R) ∪ K(B) covers at most 0.9871 · npoints of R∪B. On the other hand, drawing plane redand blue geometric spanning trees on R and B thatavoid bichromatic crossings is not always possible, andTokunaga [13] characterized their existence in termsof the bichromatic edges on CH(R ∪B).One may wonder whether using 1-plane graphswould always yield a positive solution to the aboveproblems. The answer is armative: Tokunaga [13]also proved that for given R and B, there exists a pair(path(R), path(B)) of red and blue geometric simpleHamiltonian paths such that each edge of path(R) in-tersects at most one edge of path(B) and vice versa.Having the red and blue geometric simple Hamilto-nian paths from that result, with at most one bichro-matic crossing per edge, we already have got 1-planespanning trees, and taking in each path one segmentout of any two consecutive we get a 1-plane perfectmatching with no monochromatic crossings. This 1-plane matching can also be obtained by using theHam-sandwich theorem and induction on |R ∪B|.We include this remark because the focus in [13] wasto get few crossings rather than achieving the 1-planecharacter, but the consequences show that pursuingthe latter line of research may be of interest.3 ConclusionWe have proved that several problems on bicoloredpoint sets asking for the construction of a plane geo-metric graph with some requisites, and that have ingeneral negative answer, turn out to have a solution ifthe requirement of the graphs being plane is relaxedto being 1-plane.As mentioned in a previous section, this is ongoingresearch and answering Conjecture 1 is our priority.On the other hand, we are also studying the samerelaxation for other problems in which 1-plane graphsmay provide a solution where plane graphs are notsucient.References[1] M. Abellanas, A. García, F. Hurtado, and J. Tejel,Caminos alternantes, in: Actas X Encuentros de Ge-ometría Computacional (in Spanish), 2003, 712.[2] J. Akiyama and J. Urrutia, Simple alternating pathproblem, Discr. Math. 84 (1990), 101103.[3] P. Brass, W. Moser, and J. Pach, Research Problemsin Discrete Geometry, Springer, 2005.[4] J. Cibulka, J. Kyn£l, V. Mészáros, R. Stola°, andP. Valtr, Hamiltonian alternating paths on bicol-ored double-chains, in: Graph Drawing 2008, LectureNotes in Computer Science 5417 (2009), 181192.[5] A. Dumitrescu and R. Kaye, Matching colored pointsin the plane: Some new results, Comput. Geom. The-ory Appl. 19 (2001), 6985.[6] A. Dumitrescu and W. Steiger, On a matching prob-lem in the plane, Discr. Math. 211 (2000), 183195.[7] F. Hurtado and C. D. Tóth, Plane geometric graphaugmentation: a generic perspective, Chapter 16in: Thirty Essays on Geometric Graph Theory (J.Pach, ed.), vol. 29 of Algorithms and Combinatorics,Springer, 2013, 327354.[8] A. Kaneko and M. Kano, Discrete geometry on redand blue points in the plane - a survey, in: Discreteand Computational Geometry, The Goodman-PollackFestschrift; edited by B. Aronov et al., Springer, 2003,551570.[9] A. Kaneko, M. Kano, and Y. Yoshimoto, AlternatingHamiltonian cycles with minimum number of cross-ings in the plane, Int. J. Comput. Geom. Appl. 10(2000), 7378.[10] J. Kyn£l, J. Pach, and G. Tóth, Long alternatingpaths in bicolored point sets, in: Graph Drawing2004 (J. Pach, ed.), Lecture Notes in Computer Sci-ence 3383 (2004), 340348. Also in Discr. Math. 308(2008), 43154322.[11] V. Mészáros. Extremal problems on planar point setsPhD Thesis, University of Szeged, 2011.[12] J. Pach, ed. Thirty Essays on Geometric GraphTheory, vol. 29 of Algorithms and Combinatorics,Springer, 2013.[13] S. Tokunaga, Intersection number of two connectedgeometric graphs, Information Proc. Discrete Math.150 (1996), 371378.118

The alternating path problem revisited

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The alternating path problem revisited

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