Monochromatic Plane Matchings in Bicolored Point Set

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1 CCCG 2017, Ottw, Ontrio, July 26 28, 2017 Monohromti Plne Mthings in Biolore Point Set A. Krim Au-Affsh Sujoy Bhore Pz Crmi Astrt Motivte y networks interply, we stuy the prolem of omputing monohromti plne mthings in iolore point set. Given iolore set P of n re n m lue points in the plne, where n n m re even, the gol is to ompute plne mthing M R of the re points n plne mthing M B of the lue points tht minimize the numer of rossing etween M R n M B s well s the longest ege in M R M B. In this pper, we give symptotilly tight oun on the numer of rossings etween M R n M B when the points of P re in onvex position. Moreover, we present n lgorithm tht omputes ottlenek plne mthings M R n M B, suh tht there re no rossings etween M R n M B, if suh mthings exist. For points in generl position, we present polynomil-time pproximtion lgorithm tht omputes two plne mthings with liner numer of rossings etween them. 1 Introution Let P e set of n points in the plne, where n is even. A perfet mthing of P is perfet mthing in the omplete Eulien grph inue y P. A ottlenek mthing M of P is perfet mthing of P tht minimizes the length of the longest ege of M. Let λ M enote the ottlenek of M, i.e., the length of the longest ege in M. A plne mthing is mthing with non-rossing eges. Mthing prolems hve een stuie extensively, see, e.g., [3, 5, 6, 9, 12, 13]. In this pper, we stuy the ottlenek plne mthing prolem in iolore point set. Let P e iolore set onsisting of n re n m lue points, suh tht n n m re even. Let R enote the set of the re points of P, n let B enote the set of the lue points of P. An L-monohromti mthing M of P is mthing of P, suh tht (i) M = M R M B, where M R n M B re plne perfet mthings of R n B, respetively, n (ii) λ M L. Let r(p, L) enote the minimum numer of intersetion present in ny L-monohromti Softwre Engineering Deprtment, Shmoon College of Engineering, Beer-Shev 84100, Isrel, u1@se..il. Deprtment of Computer Siene, Ben-Gurion University, Beer-Shev 84105, Isrel, sujoy.hore@gmil.om. Deprtment of Computer Siene, Ben-Gurion University, Beer-Shev 84105, Isrel, rmip@s.gu..il. The reserh is prtilly supporte y the Lynn n Willim Frnkel Center for Computer Siene. mthing of P. In this pper, we investigte the vlue r(p, L) for ifferent iolore point sets, n stuy the prolem of omputing n L-monohromti mthing. 1.1 Relte work The ottlenek plne mthing prolem for generl point set hs een first stuie y Au-Affsh et l. [2]. They showe tht the prolem is NP-hr n presente 2 10-pproximtion lgorithm. In [1], the uthors showe how to ompute plne mthing of size t lest n/5, whose eges hve length t most λ in O(n log 2 n) time, n plne mthing of size t lest 2n/5, whose eges hve length t most ( 2 + 3) λ in O(n log n) time, where λ is the length of the longest ege of ottlenek plne mthing. Crlsson n Armruster [4] prove tht the iprtite (re/lue) version of the ottlenek plne mthing prolem is NP-hr, n gve n O(n 3 log n)-time lgorithm tht solves the prolem when the points re on onvex position, n n O(n 4 log n)-time lgorithm tht solves the prolem when the re points lie on line l n the lue points lie ove (or elow) l. For iolore inputs, Merino et l. [10] otine tight oun on the numer of intersetions in monohromti minimum weight mthings for iolore point sets. Tokung [11] exmine non-rossing spnning trees of the re points n the lue points n foun tight oun on the minimum numer of intersetions etween the re n lue spnning trees. Joeris et l. [7] stuie the numer of intersetions for monohromti plnr spnning yles. In eh of the ove works, points oul hve only one of two olors n numer of intersetion points were prove to e symptotilly liner on totl numer of points. Kno et l. [8] onsiere the se of more thn two olors n stuie the numer of intersetions for monohromti spnning trees. 1.2 Our results In Setion 2, we onsier the se when the points of P re in onvex position. We give tight oun on r(p, L) in this se. Moreover, we give n lgorithm tht omputes in O( P 3 + P ) time n L- monohromti mthing of P, where L is the minimum rel numer suh tht the eges of the mthing o not ross eh other, if suh mthing exists. In Setion 3, we give polynomil-time pproximtion lgorithm for points in generl position. 7

2 29 th Cnin Conferene on Computtionl Geometry, Monohromti Mthing of Points in Convex Position In this setion, we onsier the se where the points of P = R B re in onvex position, i.e., the points of P form verties of onvex polygon. Let M R n M B enote the ottlenek plne mthings of the sets R n B, respetively. Let λ R n λ B enote the ottleneks of M R n M B, respetively, n let λ = mx{λ R, λ B }. Notie tht M R M B is not neessrily plne mthing. Notie lso tht in ny L-monohromti mthing of P, the length of the longest ege is t lest λ. Let r(p, λ) e the minimum numer of intersetion present in ny L-monohromti mthing of P with L = λ. In the following, we give tight oun of r(p, λ). 2.1 Lower oun on r(p, λ) In Figure 1, we show set of iolore points P in onvex position, suh tht the numer of rossings etween M R n M B nnot e etter thn liner. Figure 3: (, ) (, ) is mx{, }. Theorem 1 Let P = R B e set of points in onvex position. Then r(p, λ) 9k 2, where k = min{n, m}, n = R, n m = B. Proof. Assume, w.l.o.g., tht n m, i.e., k = n. For the ske of ontrition, suppose tht r(p, λ) 9n Let M P e λ-monohromti mthing of P with minimum numer of intersetions n M P = M R M B. Thus, the numer of intersetion in M P is t lest 9n Sine M R = n 2, y pigeonhole priniple there is t lest one ege (x, y) M R intersete y t lest 10 eges from M B. Assume, w.l.o.g., tht x n y re on the X-xis, n notie tht xy λ. We efine the region U = {w R 2 : (x, y) w λ}; see Figure 4. Oserve tht the totl perimeter of the region U is t most 2(π + 1)λ 8.29λ. Figure 1: Any L-monohromti mthing of P hs liner numer of rossings etween the re n the lue mthings. x y 2.2 Upper oun on r(p, λ) For n ege (, ) n point w, let (, ) w enote the minimum Eulien istne etween w n ny point on (, ); see Figure 2. For two non-interseting eges (, ) n (, ), suh tht {,,, } re in onvex position n (, ) oes not interset (, ), let (, ) (, ) enote mx{, }; see Figure 3. In the following theorem, we give n upper oun on r(p, λ). w Figure 2: (, ) w is the minimum istne etween w n ny point on (, ). w Figure 4: The set of ll points of istne t most λ from (x, y). Let E = {(p 1, q 1 ), (p 2, q 2 ),..., (p j, q j )}, where j 10, e the set of eges of M B tht interset (x, y), suh tht p 1, p 2,..., p j re ove (x, y), q 1, q 2,..., q j re elow (x, y), n, for eh 1 i < j, the intersetion point of (p i, q i ) with (x, y) is to the left of the intersetion point of (p i+1, q i+1 ) with (x, y). Lemm 2 There is t lest two eges (p i, q i ), (p i+1, q i+1 ) in E, suh tht (p i, q i ) (p i+1, q i+1 ) λ. Proof. For the ske of ontrition, suppose tht, for eh 1 i < j, we hve (p i, q i ) (p i+1, q i+1 ) > λ. Thus, for eh 1 i < j, either p i p i+1 > λ or q i q i+1 > λ. Therefore, the totl perimeter of the onvex polygon S forme y {p 1, p 2,... p j, q j, q j 1,..., q 1 } is t lest j 1 p i p i+1 + q i q i+1 > 9λ, sine j 10. i=1 This ontrits the ft tht S is ontine insie the onvex region U whose perimeter is t most 8.29λ. 8

3 CCCG 2017, Ottw, Ontrio, July 26 28, 2017 By Lemm 2, there re two jent eges (p i, q i ), (p i+1, q i+1 ) E, suh tht p i p i+1 λ n q i q i+1 λ. We reple the eges (p i, q i ) n (p i+1, q i+1 ) in M B y the eges (p i, p i+1 ) n (q i, q i+1 ) to otin new ottlenek plne mthing M B of B of ottlenek t most λ. Let M P = M R M B. Clerly, M P is λ-monohromti mthing of P. In the following lemm, we show tht the new eges (p i, p i+1 ) n (q i, q i+1 ) o not inrese the numer of intersetions in M P. Lemm 3 If (p i, p i+1 ) or (q i, q i+1 ) intersets n ege (u, v) in M R, then either (p i, q i ) or (p i+1, q i+1 ) intersets (u, v) in M P. Proof. Assume, w.l.o.g, tht (p i, p i+1 ) intersets n ege (u, v) in M R. Let Q e the qurngle otine y points x, p i, p i+1 n y. Clerly, Q is empty of points of P, sine P in onvex position. Moreover, sine the eges of M R o not interset eh other, (u, v) n (x, y) o not interset. Thus, (u, v) intersets (p i, q i ) or (p i+1, q i+1 ) ; see Figure 5. v x p i u p i+1 q i q i+1 Figure 5: An illustrtion of Lemm 3. By Lemm 3, the eges (p i, p i+1 ) n (q i, q i+1 ) o not inrese the numer of intersetions in M P. However, (p i, p i+1 ) n (q i, q i+1 ) erese the numer of intersetions in M P, sine they o not interset (x, y). This implies tht the numer of intersetions in M P is less thn tht in M P, whih ontrits the ssumption tht M P is λ-monohromti mthing with minimum numer of intersetions. This ompletes the proof of Theorem 1. Remrk. Au-ffsh et l. [1] ompute ottlenek plne mthing of set of points in onvex position in O( P 3 ) time. We use their lgorithm to ompute ottlenek plne mthings of R n of B, seprtely. Then in itionl O( P ) time, we ompute λ-monohromti mthing M P of P y onsiering eh ege (x, y) of M R seprtely, n if there re two jent eges (p i, q i ), (p i+1, q i+1 ) M B interseting (x, y), suh tht p i p i+1 λ n q i q i+1 λ, then we reple them y the eges (p i, p i+1 ) n (q i, q i+1 ). By the ove lemms, the numer of intersetions in M P is t most 9k 2, where k = min{n, m}. y 2.3 Monohromti plne mthing Let L e the minimum vlue suh tht there exists n L-monohromti mthing of P with no rossings, if suh mthing exists. Tht is, L is the minimum vlue suh tht r(p, L) = 0, if suh vlue exists. An optiml plne mthing of P is n L-monohromti mthing of P with no rossings. In this setion, we present polynomil-time lgorithm tht omputes n optiml plne mthing of P, if suh mthing exists, Let P = R B e set of points in onvex position, where R ontins n even numer of re points n B ontins m even numer of lue points. Rell tht M R (resp., M B ) is ottlenek plne mthing of R (resp., B) of ottlenek λ R (resp., λ B ) n λ = mx{λ R, λ B }. Thus, the length of the longest ege in ny L-monohromti mthing of P is t lest λ. Our lgorithm omputes the minimum vlue L, suh tht r(p, L) = 0 (if exists), n onstruts n optiml plne mthing of P. Let us ssume tht R = {r 1,..., r n }, B = { 1,..., m }, n P = {p 1,..., p n+m }, suh tht p i R B. Let P = R B enote the verties of the onvex polygon, n re orere in lokwise orer; see Figure 6. Assume, w.l.o.g., tht p 1 = r 1. Notie tht, ottlenek plne mthing M R of R n ottlenek plne mthing M B of B n e ompute seprtely in polynomil time [2]. However, the eges of M R my ross the eges of M B. Let M P enote n optiml plne mthing of P. We first stte the following oservtion. Oservtion 1 For eh ege (p i, p j ) in M P, (i) p i n p j hve the sme olor, (ii) (p i, p j ) prtitions the points of P into two isjoint point sets, suh tht the numer of re points n lue points ontine in eh set is even, n (iii) i + j is o; see Figure 6. p 12 r 1 r 1 p p 10 6 r 5 r 2 3 p 5 p 4 p 5 9 r 4 r 3 4 p 6 p 8 Figure 6: The onvex polygon tht is otine from P. p 1 n e mthe to the points p 4, p 8, p 12. Bse on this oservtion, we efine weight funtion for eh pir of points in P s follows. For eh 1 i < j n+m, we efine w i,j = p i p j, if p i n p j re of the sme olor, i + j is o, n the ege (p i, p j ) prtitions the points in P into two isjoint point sets suh tht the numer of re points n lue points ontine in eh p 1 p 7 p 2 p 3 9

4 29 th Cnin Conferene on Computtionl Geometry, 2017 set is even. Otherwise, w i,j =. Let P [i, j] enote the set of points {p i, p i+1,..., p j }. Let l i,j e the minimum vlue, suh tht r(p [i, j], l i,j ) = 0. Hene, L = l 1,n+m. Let p k e the point mthe to p 1 in M P. Thus, L = l 1,n+m = mx{w 1,k, l 2,k 1, l k+1,n+m }. Therefore, to ompute l 1,n+m, we ompute mx{w 1,k, l 2,k 1, l k+1,n+m }, for eh even k etween 2 n n + m, n tke the minimum over these vlues. In generl, for eh 1 i < j n + m, suh tht i + j is o, we ompute l i,j y w i,k, if k = i + 1 = j mx{w i,k, l k+1,j }, if k = i + 1 min k=i+1,i+3,,j mx{w i,k, l i+1,k 1 }, if k = j mx{w i,k, l i+1,k 1, l k+1,j }, otherwise. We ompute L = l 1,n+m using ynmi progrmming. The ynmi progrmming tle T ontins (n + m) rows n (n + m) olumns n the entry T [i, j] orrespons to solution of the prolem for the set P [i, j]. Notie tht, eh entry T [i, j] is ompute y proessing O(n + m) entries tht re lrey ompute. Hene, we n ompute L = l i,j = T [1, n + m] in O((n + m) 3 ) time. Thus, we hve the following theorem. Theorem 4 Given set P of n re n m lue points in onvex position, where n n m re even, one n ompute in O((n+m) 3 ) time the minimum vlue L, suh tht r(p, L) = 0, n if L is finite, then n optiml plne mthing n e ompute in O((n + m) 3 ) time. 3 Monohromti Mthing of Points in Generl Position Let P = R B e set of iolore points in the plne, suh tht R ontins n re points, B ontins m lue points, n n n m re even. In this setion, we present polynomil-time pproximtion lgorithm tht omputes n L-monohromti mthing of P with t most O( P ) rossings. Let M R n M B e ottlenek plne mthings of R n B, respetively. Let λ R n λ B e the ottleneks of M R n M B, respetively, n let λ = mx{λ R, λ B }. Rell tht, in ny L-monohromti mthing of P, the length of the longest ege is t lest λ. In [2], Auffsh et l. prove tht omputing ottlenek plne mthing of set of points in generl position in the plne is NP-Hr. This implies tht it is NP-Hr to ompute λ-monohromti mthing of P sine λ = mx{λ R, λ B }. However, in this setion we give polynomil time pproximtion lgorithm tht omputes (2 10λ)-monohromti mthing of P with liner numer of intersetions. Tht is, we ompute n L-monohromti mthing M P with L = 2 10λ, suh tht totl numer of intersetions etween the re n the lue eges in M P is O( P ). Let MR n M B e ottlenek mthings of R n B tht my hve rossings, respetively. Let λ R n λ B e the ottleneks of MR n M B, respetively. Sine M R n MB n e ompute in polynomil-time [5], we ssume, w.l.o.g., tht λ B λ R. Thus, sine λ B λ B n λ R λ R, we hve λ R λ. In the rest of this setion, we prove the following theorem. Theorem 5 Let P = R B e set of iolore points in the plne, suh tht ontins n re points, B ontins m lue points, n n n m re even. Then, there is polynomil-time pproximtion lgorithm tht omputes (2 10λ R )-monohromti mthing M P of P, suh tht the numer of intersetions in M P is O( P ). Proof. Let MP e M R M B. We egin y lying gri of sie length 2 2λ R. Assume, w.l.o.g. tht no point of P lies on the ounry of gri ell. Eh ege of MP is either ontine in gri ell or onnets two points from two jent ells (i.e., two ells shring sie or orner). For n ege e in MP, we sy tht e is n internl ege if it is ontine in gri ell, n n externl ege otherwise. An externl ege n e of two types: stright externl ege (s-ege for short) onnets etween two points in two gri ells tht shre sie, n igonl externl ege (-ege for short) onnets etween two points in two gri ells tht shre orner. Let C e gri ell. The egree of C is enote y eg(c) n it is equl to the numer of externl eges of MP with n enpoint in C. Our lgorithm onsists of two stges. In Stge 1, we onvert MP into new mthing M P of P, suh tht eg(c) 8, for eh gri ell C, n, in Stge 2, we onstrut (2 10λ R )-monohromti mthing M P se on M P. Stge 1 In this stge, whih is tken from [2], we onvert MR (resp., MB ) to new perfet mthing M R (resp., M B ) of R (resp., of B). The onversion is one y pplying sequene of rules on MR n on M B seprtely. Eh rule is pplie s long s there is n instne in the urrent mthing to whih it n e pplie. When there re no more suh instnes, we move to the next rule in the sequene. Rule 1: If there re two -eges (, ) n (, ) ssoite with the sme orner, suh tht n in the sme ell n n in the sme ell, then these eges re reple y two internl eges (, ) n (, ); see Figure 7(). Rule 2: If there re two -eges (, ) n (, ) ssoite with the sme orner, suh tht,,, n re in ifferent ells, then these eges re reple y two s-eges (, ) n (, ); see Figure 7(). For -ege (, ) tht onnets etween two ells C n C n ssoite with orner r, we efine 10

5 CCCG 2017, Ottw, Ontrio, July 26 28, 2017 () Figure 7: () Two -eges (, ) n (, ) re reple y two internl eges (, ) n (, ), n () two - eges (, ) n (, ) re reple y two s-eges. nger zone of (, ) in eh of the two other ells shring r s n isoseles right tringle; see Figure 8(). The length of its sies is 2λ R n it is semi-open (i.e., it oes not inlue the hypotenuse of length 2λ R ). () reple (, ) n (, ) y two s-eges (, ) n (, ). Rule 4: This rule is pplie on -ege (, ) n n s-ege (, ), suh tht n re in the sme ell, n n re in two jent ells tht shre sie; see Figure 9(). We reple (, ) n (, ) y n internl ege (, ) n n s-ege (, ). Rule 5: This rule is pplie on two s-eges (, ) n (, ), suh tht n re in the sme ell, n n re in the sme ell; see Figure 9(). We reple (, ) n (, ) y two internl ege (, ) n (, ). C r C 2 C C r C 2 C () () Figure 9: () Rule 4, n () Rule 5. C () r r C () C 2 C C 2 r C Lemm 6 (Lemm 3.2 in [2]) Let M P e M R M B. Then, M P hs the following properties. 1. An ege is either ontine in single ell, or onnets etween pir of points in two jent ells. 2. A orner of the gri hs t most one -ege of M R n t most one -ege of M B ssoite with it. 3. A -ege in M P is of length t most λ R. () Figure 8: () The nger zone efine y the ege (, ) in n C 2, () (, ) is n s-ege, () (, ) is n internl ege n is not in nother nger zone in, n () (, ) is n internl ege n is in nother nger zone in. Rule 3: This rule is pplie on -ege (, ) n n ege (, ) with n enpoint insie nger zone efine y (, ); see Figure 8( ). 0 If (, ) is n s-ege, then its other enpoint is in one of the ells or C 2 ; see Figure 8(). In this se, we reple (, ) n (, ) y n internl ege (, ) n n s-ege (, ). 0 If (, ) is n internl ege, then onsier the other enpoint of (, ). If is not in nger zone in efine y nother -ege, then we reple (, ) n (, ) y two s-eges (, ) n (, ); see Figure 8(). If is in nger zone in efine y nother -ege (, ), then (, ) is ssoite with one of the two orners of jent to the orner r; see Figure 8(). Therefore, either C or C ontins n enpoint of oth (, ) n (, ). We () 4. The two nger zones efine y -ege of M R (resp., M B ) re empty of points of R (resp., of B). 5. Eh ell C ontins t most 4 externl eges of M R n 4 externl eges of M B, n eg(c) If -ege in M R (resp., in M B ) onneting etween ells n C 2, n C is ell shring sie with oth n C 2, then there is no s-ege in M R (resp., in M B ) onneting etween C n either or C 2. Stge 2 In this stge, we onstrut (2 10λ R )-monohromti mthing M P of P se on M P = M R M B. We onsier eh ell seprtely. Let C e non-empty gri ell n let R C (resp., B C ) e the set of points of R (resp., B) lying in C. Rell tht there re t most 4 externl eges of M R n t most 4 externl eges of M B ssoite with C. We use the sme proeure of [2] to selet the points in R C (resp., in B C ) tht will serve s enpoints of externl eges of M R (resp., of M B ), suh tht the externl eges of M R (resp., of M B ) tht will e onnete to these point o not ross eh other. For eh externl ege e of M R (resp., of 11

6 29 th Cnin Conferene on Computtionl Geometry, 2017 M B ) onneting etween two ells n C 2, let n e the points tht were hosen s the enpoint of e in n C 2, respetively. We the ege (, ) to M R (resp., to M B ). It hs e prove in [2] tht the length of (, ) is t most 2 10λ R. Let RC E R C (resp., BC E B C) e the set of points of R C (resp., of B C ) tht were hosen s enpoints of externl eges, n let RC I = R C \ RC E (resp., BI C = B C \BC E). By the wy we selet the points of RE C (resp., of BC E), the points in RI C (resp., BI C ) re ontine in onvex polygon X R C (resp., X B C), suh tht ny externl ege of M R (resp., M B ) with enpoint in RC E (resp., in RE C ) oes not interset the interior of X R (resp., X B ). For eh ell C, we ompute minimum weight mthing M(RC I ) (resp., M(RI C )) of the points in RC I (resp., in BI C ) n we it to M R (resp., to M B ). Clerly, the eges of M(RC I ) o not ross eh other n o not ross the externl eges tht re onnete to points from RC E, n the eges of M(BI C ) o not ross eh other n o not ross the externl eges tht re onnete to points from BC E. Moreover, the length of eh ege in M(RC I ) n in M(BI C ) is t most 4λ R. Let M P e M R M B. Sine M R (resp., M B ) is plne mthing n eh ege in M R (resp., in M B ) is of length t most 2 10λ R, M P is (2 10λ R )- monohromti mthing of P. We now show tht the numer of intersetions in M P, i.e., etween the eges of M R n the eges of M B, is O( P ). We oun the numer of intersetions for eh ell seprtely. In eh ell C, there woul e three types of intersetions: intersetion etween n externl ege of M R n n externl ege of M B, intersetion etween n externl ege of M R n n internl ege of M B (n vie vers), n intersetion etween n internl ege of M R n n internl ege of M B. We oun the numer of intersetion of eh type seprtely. Rell tht the numer of externl eges of M R (resp., of M B ) tht hve one enpoint in RC E (resp., in BE C ) is t most 4. This implies tht eh externl ege of M R n e intersete y t most 4 externl eges of M B, n thus the totl numer of intersetions etween the externl eges of M P tht hve enpoint in C is t most 4 RC E. Moreover, the totl numer of intersetions etween the externl eges of M P tht hve enpoint in C n the internl eges of M(RC I ) M(BI C ) is t most 2( RC I + BI C ). Finlly, the totl numer of intersetions etween the internl eges of M(RC I ) n the internl eges of M(BC I ) is t most ( RI C + BI C )/2, y Theorem 1 in [10]. Thus, the numer of intersetions proue y the points in C is t most 4( R C + B C ). Therefore, the numer of intersetions in M P is t most 4( R + B ) = 4 P. This ompletes the proof. Referenes [1] A. K. Au-Affsh, A. Biniz, P. Crmi, A. Mheshwri, n M. H. M. Smi. Approximting the ottlenek plne perfet mthing of point set. Comput. Geom., 48(9): , [2] A. K. Au-Affsh, P. Crmi, M. J. Ktz, n Y. Trelsi. Bottlenek non-rossing mthing in the plne. Comput. Geom., 47(3): , [3] G. Aloupis, J. Crinl, S. Collette, E. D. Demine, M. L. Demine, M. Dulieu, R. Fil-Monroy, V. Hrt, F. Hurto, S. Lngermn, M. Sumell, C. Ser, n P. Tslkin. Mthing points with things. In LATIN, volume 6034 of LNCS, pges , [4] J. G. Crlsson, B. Armruster, S. Rhul, n H. Bellm. A ottlenek mthing prolem with ege-rossing onstrints. Int. J. Comput. Geometry Appl., 25(4): , [5] M. S. Chng, C. Y. Tng, n R. C. T. Lee. Solving the eulien ottlenek mthing prolem y k- reltive neighorhoo grphs. Algorithmi, 8(1 6): , [6] A. Efrt, A. Iti, n M. J. Ktz. Geometry helps in ottlenek mthing n relte prolems. Algorithmi, 31(1):1 28, [7] B. Joeris, I. Urruti, n J. Urruti. Geometri spnning yles in ihromti point sets. Grphs n Comintoris, 31(2): , [8] M. Kno, C. Merino, n J. Urruti. On plne spnning trees n yles of multiolore point sets with few intersetions. Inf. Proess. Lett., 93(6): , [9] L. Lovász n M. D. Plummer. Mthing Theory. Elsevier Siene Lt, [10] C. Merino, G. Slzr, n J. Urruti. On the intersetion numer of mthings n minimum weight perfet mthings of multiolore point sets. Grphs n Comintoris, 21(3): , [11] S. Tokung. Intersetion numer of two onnete geometri grphs. Inf. Proess. Lett., 59(6): , [12] P. M. Viy. Geometry helps in mthing. SIAM J. Comput., 18(6): , [13] K. R. Vrrjn. A ivie-n-onquer lgorithm for min-ost perfet mthing in the plne. In FOCS, pges ,

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