On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W prov that for ach 1 i k w can find a plan spanning tr T i of P i such that th dgs of T 1,..., T k intrsct at most kn(k 1)(n k)+ (k)(k 1), whr n is th numbr of points in P 1... P k. If th intrsction of th convx hulls of P 1,..., P k is non mpty, w can find k spanning cycls such that thir dgs intrsct at most (k 1)n tims, this bound is tight. W also prov that if P and Q ar disjoint point sts in gnral position, thn th minimum wight spanning trs of P and Q intrsct at most n tims, whr P Q = n (th wight of an dg is its lngth). 1 Introduction Th study of gomtric graphs, that is graphs whos vrtx st is a collction of points on th plan in gnral position and its dgs ar straight lin sgmnts conncting pairs of vrtics, has rcivd a lot of attntion latly. Numrous problms in which w want to draw graphs on th plan such that thir vrtics li on th lmnts of a fixd point st hav bn studid. Ramsy typ problms in which w want to color th dgs or vrtics of a gomtric graph such that som spcific forbiddn subgraphs do not appar hav also bn studid. Th intrstd radr may consult a rcnt survy by J. Pach [9] containing many rsults in this fild. In this papr w ar intrstd in studying problms of mbddings of gomtric graphs on colord point sts. Ths problms hav bn studid for som tim now, for xampl in [1, ] th problm of mbdding trs and altrnating paths on bicolord point sts is studid. In [,, 5] matching problms on colord point sts ar studid. For two colord point sts w ar intrstd in obtaining matchings in which vry dg has its Dpartmnt of Computr and Information Scincs, Ibaraki Univrsity, Hitachi 16-511, Japan Instituto d Matmáticas Univrsidad Nacionál Autónoma d México, México D.F. México 1
ndpoints of diffrnt (or qual) color. A wll known rsult stats that givn a colction P n of n points in gnral position, n blu, and n rd, w can always match a blu with a rd point in P n such that th lin sgmnts joining matchd pairs of points do not intrsct. For a rcnt survy daling with numrous problms on colord point sts s [6]. Problms in which instad of coloring th vrtics, w color th dgs of gomtric graphs, ar studid in [, ]. Lt P n b a st with n points on th plan. A spanning tr of P n is a connctd gomtric graph with vrtx st P n containing xactly n 1 dgs. Lt P and Q b disjoint point st. Tokunaga, studid and solvd th problm of finding spanning trs for P and Q with th smallst possibl numbr of dg intrsctions. It turns out that this numbr dpnds only in th ordr in which th lmnts of P and Q lying on th convx hull Conv(P Q) of P Q appar. Mor spcifically, lt p 0,..., p r 1 b th points on Conv(P Q) in clockwis ordr, and lt i b th numbr of indxs j such that p j and p j+1 ar on in P and th othr in Q, addition takn mod r. Thn it is always possibl to find spanning trs for P and Q such that thir dgs do not intrsct if i, othrwis thy intrsct xactly i tims. This implis for xampl that if all th points on th convx hull ar th sam color, or all th points in P which blong to th convx hull appar in conscutiv ordr, thn w can find spanning trs for P and Q which do not intrsct, rgardlss of how many or whr th rmaining points of P and Q ar. In this papr w study th following problm: Lt P 1,..., P k b a family of disjoint point sts such that P 1... P k is in gnral position. Find spanning trs for P 1,..., P k such that thir dgs hav as fw intrsctions as possibl. In this papr w prov th following rsult: Thorm 1 Lt P 1,..., P k b a collction of disjoint point sts. Thn w can find for ach P i a spanning tr T i such that th total numbr of intrsctions among th dgs of T 1,..., T k is at most (k 1)(n k) + (k)(k 1) whr P 1... P k = n. This bound is tight within a factor of two from th optimal solution. W also giv similar rsults for spanning cycls of familis of point sts P 1,..., P k in which Conv(P 1 )... Conv(P k ) is non mpty. Sharp bounds for this problm ar obtaind. In th last sction of this papr, w prov th following rsult that is of indpndnt intrst: Lt P and Q b disjoint point sts, thn thir uclidan minimum wight spanning trs intrsct at most 1n tims. Using this w prov th following rsult: Lt P 1,..., P k b familis of disjoint point sts such that P 1... P k is in gnral position. For ach P i lt T i b its uclidan minimum wight spanning tr, i = 1,..., k. Thn thn th dgs of ths trs intrsct at most kn tims.
Spanning trs with fw intrsctions Givn two disjoint point sts P 1 and P it is not always possibl to find spanning trs for thm such that thir dgs do not intrsct. In fact if w hav s points which ar th vrtics of a convx polygon such that altrnatly thy blong to P 1 and P, thn it is asy to vrify that any spanning tr for P 1 intrscts any spanning tr for P at last s 1 tims, s Figur 1. From hr th following obsrvation follows: Obsrvation 1 Thr ar familis of point sts P 1,..., P k with P 1,..., P k = sk such that thir dgs intrsct at last k(k 1) (s 1) tims. Figur 1: Two sts of points, ach with six points such that any spanning tr of th st with solid points intrscts any spanning tr for th rmaining points at last fiv tims. If w considr a similar problm for thr or mor point sts our problm bcoms much hardr, vn for points in convx position. Lt P b a st of n = ks points in convx position lablld p 1,..., p sk. Split P into k substs P 1,..., P k such that th lmnt p i+rk blongs to P i, r = 0,..., s 1. Finding for ach P i a spanning tr T i, 1 i k, such such that thir dgs hav th fwst possibl numbr of intrsctions is hard. W now show a st of spanning trs T 1,..., T k such that thir dgs intrsct at most ( k k)(s 1) k(k ) tims if k is vn; othrwis thy intrsct ( (k 1) + k 1 (k 1) )(s 1) tims, i.. th numbr of tims thir dgs intrsct is at most tims th optimal solution. For i vn, lt T i b th tr containing th dgs joining p i+ak to p i+bk, a + b = s + 1 or a + b = s +, 1 a, b s. For i odd, T i, is th tr containing th dgs joining p i+ak to p i+bk, a + b = s or a + b = s + 1, 1 a, b s. Notic that two trs T i and T j intrsct s 1 tims if i and j hav diffrnt parity; othrwis thy intrsct (s 1) 1 = s tims. S Figur. Thrfor ths trs intrsct xactly ( k k)(s 1) k(k ) if k is vn; and ( (k) + k 1 (k 1) )(s 1) if k is odd. Morovr w bliv that this configuration is, in fact, clos to th optimal solution for point sts in convx position.
1 5 6 1 5 6 9 Figur : On th lft hand sid w hav T i, i odd, with dashd lins and T j, j vn, with solid lins; s = so, thy intrsct tims. On th right hand sid w hav T i and T j, i, j vn; as s = 9, thy intrsct 15 tims. W procd now to study our problm for point sts in gnral position. Suppos w.l.o.g. that th points in P 1... P k hav diffrnt x-coordinats, and P i, i = 1,..., k. Assum that for vry i th lmnts of P i ar labld p i,1, p i,,..., p i,ri such that if r < s thn th x-coordinat of p i,r is smallr than th x-coordinat of p i,s. Lt T i b th path with vrtx st P i in which p i,j is connctd to p i,j+1 by an dg dnotd by i,j, j = 1,..., r i 1. S Figur. Figur : A colction of four point sts and thir spanning trs. Th point sts ar th vrtics of our trs, which turn out to b paths. Lmma 1 Th dgs of T i and T j intrsct at most r i + r j tims. Proof: Our rsult is clarly tru if r i + r j, or on of T i or T j has xactly on dg. Suppos now that th x coordinat of p i, is smallr than that of p j,. Thn th dg i,1 of T i joining p i,1 to p i, intrscts at most on dg of T j, namly th dg j,1 joining p j,1 to p j,. Rmov p i,1 from p i, and by induction our rsult follows. In a similar way w can prov:
Lmma Th dgs of T 1,..., T k intrsct at most (k 1)(n k) + (k)(k 1) tims, whr P 1... P k = n Proof: Our rsult is tru if T 1,..., T k hav togthr at most k dgs, in fact in this cas if all of thm intrsct ach othr, thir total numbr of intrsctions is (k)(k 1). Suppos thn that our trs contain mor than k dgs, and lt i,1 b such that th x-coordinat of p i, is smallr than th x-coordinat of p j,, i j, 1 j k. Thn th dg i,1 joining p i,1 to p i, intrscts at most k 1 dgs, i.. in ach T j i,1 intrscts at most th dg j,1 joining p j,1 to p j,, j i. Rmoving this dg, and p i1 from P i, and procding by induction on P 1,..., P i {p i,1 },..., P k our rsult follows. Obsrv that th bound dtrmind in Lmma is within a factor of two of that in Obsrvation 1. Thorm 1 follows dirctly from Obsrvation 1 and Lmma. Spanning Cycls W now study th following problm: Lt P 1,..., P k b a family of disjoint point sts such that Conv(P 1 ),..., Conv(P k ). Find a family of spanning cycls C i,..., C k of P 1,..., P k rspctivly with fw intrsctions. W prov: Thorm Lt P 1,..., P k b a family of disjoint point st such that Conv(P 1 ),..., Conv(P k ). Thn for ach P i w can find a cycl C k which covrs th vrtics of P i such that th dgs of all cycls C i,..., C k intrsct at most (k 1)n tims. Our bound is optimal. Proof: Lt q b a point in th intrior of Conv(P 1 ),..., Conv(P k ). For ach P i dfin a cycl C q i as follows: Sort th lmnts of P i around q in th countrclockwis ordr according to thir slop and labl thm p i,1,..., p i,ri (s Figur (a)). A straightforward modification to our counting argumnt in Lmma shows that th dgs of C 1,..., C k intrsct at most (k 1)n tims. To show that our bound is tight, choos n = kr, and choos kr points on a unit circl labld p 1,..., p kr, and lt P i = {p i+ks : k = 0,..., r 1}. It is asy to s that th (uniqu) cycls C i that covr th vrtics of ach P i, i = 1,..., k intrsct (k 1)(kr) = (k 1)n tims, s Figur (b). 5
p i, p i,1 q p i,ri (a) (b) Figur : Minimum wight spanning trs Th uclidan minimum wight spanning tr of a point st P n is a tr with vrtx st P n such that th sum of th lngths of its dgs is minimizd. In this sction w prov that if P 1,..., P k ar disjoint point sts and T 1,..., T k ar thir corrsponding uclidan minimum wight spanning trs thn th total numbr of intrsctions among thir dgs is at most (k 1)(n k) whr P 1... P k = n. Our proof is basd on th following obsrvation that is asy to prov: Lt T i and T j b th minimum wight spanning trs of P i and P j. Lt b any dg of T i. Thn thr is a constant c such that intrscts at most c dgs of T j whos lngth is gratr than or qual to th lngth of. It follows that th dgs of T i and T j intrsct a linar numbr of tims. In fact, w can prov that c is at most 9, howvr th proof is long, tdious, and unnlightning. W skip th dtails, thy can b supplid by th authors upon rqust. Summarizing w hav: Lmma Lt T 1 and T b th minimum wight spanning trs of two point sts P 1 and P, and any dg of T 1. Thn intrscts at most 9 dgs of T whos lngth is grathr than or qual to th lnght of. W now prov: Thorm Lt T 1,..., T k b rspctivly th minimum wight spanning trs of k point sts P 1,..., P k such that P 1... P k = n. Thn th dgs of T 1,..., T k intrsct at most 9(k 1)(n k) tims. 6
Proof: Obsrv first that sinc P 1... P k = n, T 1,..., T k hav xactly n k dgs. Lt us construct th intrsction graph H of th st of dgs of T 1,..., T k, that is th graph whos vrtx st is th st of all dgs of T 1,..., T k, two of which ar adjacnt if thy intrsct. Orint th dgs of this graph as follows: If two dgs T i and T j intsct and is longr than orint th dg in H joining thm from to, s Figur 5. 1 1 5 6 6 5 (a) (b) Figur 5: Th intrsction graph of thr minimum wight spanning trs. By Lmma 6 vry dg in T i intrscts at most 9 dgs in ach T j, i j which ar th sam lnght or longr than itslf. Thus th out-dgr of ach vrtx of H is at most 9(k 1). Our rsult follows. Obsrv that for th cas whn w hav two point sts P and Q such that P Q = n, our prvious rsults implis that th dgs of thir minimum wight spanning trs intrsct at most 9(n ) tims. This bound is far from optimal. In fact w hav bn unabl to produc xampls in which th minimum wight spanning trs of P and Q intrsct mor than n tims. An xampl is constructd as follows: P consists of points r, s, t such that r and s ar quidistant from t, and th angl rts is slightly biggr than π. Th points of Q li on a zig-zag polygonal such that ach scond sgmnt of it is paralll, and th angl btwn two conscutiv sgmnts is π as shown in Figur 6. W conclud by posing th following qustion: Opn problm 1 Is it tru that th dgs of th minimum wight spanning trs of any two point sts P and Q such that P Q = n intrsct at most n c tims, c a constant?
....... Figur 6: P has points, and Q 5. Th numbr of lmnts of Q can b incrasd to n, n. Th numbr of dg intrsctions of th minimum wight spanning trs of P and Q is n. Rfrncs [1] M. Abllanas, J. García, G. Hrnándz, N. Noy, and P. Ramos, Bipartit mbddings of trs in th plan, in Graph Drawing 96, Lctur Nots in Computr Scinc, 1990 Springr Vrlag, Brlin, (199), 1-10. [] J. Akiyama, and J. Urrutia, Simpl altrnating path problms, Discrt Math (1990) 101-10. [] A. Dumitrscu, and J. Pach. Partitioning colord point sts into monochromatic parts, Int. J. of Computational Gomtry and Applications. [] A. Dumitrscu, and R. Kay, Matching colord points in th plan, som nw rsult, Comp. Gomtry, Thory and Applications 19 (1) (001), 69-5. [5] A. Dumitrscu, and W. Stigr, On a matching problm on th plan, Discrt Math. 11 (000), 1-195. [6] A. Kanko M. Kano, Som problms and rsults on graph thory and discrt gomtry, 001. (ps-fil in http://gorogoro.cis.ibaraki.ac.jp/) [] G. Karoli, J. Pach, and G. Tóth, Ramsy-typ rsults for gomtric graphs, I, Discrt and Comp. Gomtry 1, -55 (1995). [] G. Karoli, J. Pach, and G. Tóth, and P. Valtr, Ramsy-typ rsults for gomtric graphs, II, Discrt and Comp. Gomtry 0, 5- (199).
[9] J. Pach, Gomtric graph thory. Survys in combinatorics, 1999 (Cantrbury), 16 00, London Math. Soc. Lctur Not Sr., 6, Cambridg Univ. Prss, Cambridg, 1999. [10] S. Tokunaga, Crossing numbr of two-connctd gomtric graphs, Info. Proc. Lttrs 59 (1996) 1-. 9