On Lol Trnsformtions in Pln Gomtri Grphs Em on Smll Gris Mnul Allns Prosnjit Bos Alfro Grí Frrn Hurto Pro Rmos Euro Rivr-Cmpo Jvir Tjl Astrt Givn two n-vrtx pln grphs G 1 = (V 1, E 1 ) n G 2 = (V 2, E 2 ) with E 1 = E 2 m in th n n gri, with stright-lin sgmnts s gs, w show tht with squn of O(n) point movs (ll point movs sty within 5n 5n gri) n O(n 2 ) g movs, w n trnsform th ming of G 1 into th ming of G 2. In th s of n-vrtx trs, w n prform th trnsformtion with O(n) point n g movs with ll movs stying in th n n gri. W prov tht this is optiml in th worst s s thr xist pirs of trs tht rquir Ω(n) point n g movs. W lso stuy th quivlnt prolms in th ll stting. 1 Introution Informlly, lol trnsformtion is n oprtion prform on th vrtis n gs of grph. Th trm lol is us us gnrlly th oprtion os not fft th whol grph. Typilly, th vrtis of th grph fft y lol trnsformtion r th nighorhoo of onstnt numr of vrtis. For xmpl, n g ontrtion is lol trnsformtion tht ffts th nighorhoo of two vrtis. On lol trnsformtion hs n fin, proprtis n pplitions of th lol trnsformtion with rspt to givn lss of grphs r stui [3 6, 9 12, 14, 16 20, 23]. A nturl qustion with rspt to lol trnsformtions is: Fult Informáti, U. Politéni Mri, Mri, Spin Shool of Computr Sin, Crlton U., Ottw, Cn. F. Cinis. Dp. Métoos Estístios, U. Zrgoz, Zrgoz, Spin Dprtmnt Mtmàti Apli II, U. Politèni Ctluny, Brlon, Spin Dp. Mtmátis, U. Allá, Mri, Spin Dp. Mtmátis, U. Autónom Mtropolitn-Iztplp, Méxio D.F., Méxio This work ws initit whn th uthors wr ttning workshop t th Univrsi Zrgoz. Th son n sixth uthors wr on stil lv t UPC. This work is prtilly support y MCYT TIC02-4486-C02-1, SAB 2000-0234 grnt of MECD Spin, grnt y Conyt Mxio, PIV 2001 grnt of Gnrlitt Ctluny, NSERC Cn, MCYT-FEDER BFM2002-0557, MCYT-FEDERBFM2003-0368, Gn. Ct 2001SGR00224, n DGA-2002-22861. As thr r mny iffrnt flvors of g ontrtions, for this xmpl, w ssum tht n g ontrtion in simpl grph is th oprtion of lting n g n mrging th two npoints of th g into nw vrtx whos nighorhoo is th union of th nighorhoos of th two npoints. All multipl gs r rmov so tht th rsulting grph is still simpl. 1
Dos prforming lol trnsformtion of grph in givn lss kp th grph in th sm lss? For xmpl, if th lss is iprtit grphs, thn prforming n g ontrtion on iprtit grph os not nssrily kp it iprtit. Howvr, if th lss of grphs is omplt grphs, thn g ontrtions o kp th grph in th sm lss. Anothr qustion tht is oftn stui is: Givn n instn of grph of prtiulr lss, n ll th grphs of this lss numrt vi lol trnsformtions on th givn instn? Th lol trnsformtion tht initit this stuy is rfrr to s n g flip or mor gnrlly n g mov. Th lss of grphs on whih g flips r fin is tringultions (n somtims nr-tringultions ) n g movs r normlly fin on plnr grphs. Th oprtion of n g flip or g mov is simply th ltion of n g, follow y th insrtion of nothr g suh tht th rsulting grph rmins plnr n simpl. Mor forml finitions r givn in th nxt stion. Wgnr [22] prov tht givn ny two tringultions G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) with V 1 = V 2 = n, thr lwys xists finit squn of g movs tht trnsforms G 1 into grph G 3 = (V 3,E 3 ) tht is isomorphi to G 2. Tht is, thr xists mpping φ : V 2 V 3 suh tht for u,v V 2, uv E 2 if n only if φ(u)φ(v) E 3. Susquntly, Komuro [13] show tht in ft O(n) g flips suffi. Rntly, Bos t l. [2] show tht O(log n) simultnous g flips suffi n r somtims nssry. This stting of th prolm is rfrr to s th omintoril stting sin th tringultions r only m omintorilly, i.. th yli orr of gs roun h vrtx is fin. Originl grph Invli gomtri flip Vli omintoril flip Dlt g n g. This is vli omintoril g mov sin th grph is still plnr ut it is n invli gomtri g mov sin th gs n intrst proprly. Figur 1: Vli omintoril g flip ut invli gomtri g flip. In th gomtri stting, tringultion or nr-tringultion is m in th pln suh tht th vrtis r points n th gs r stright-lin sgmnts. Hnforth, w only onsir grphs m in th pln hving stright-lin sgmnts for gs. Eg flips n g movs r still vli oprtions in this stting, xpt tht now th g tht is must A nr tringultion is pln grph whr vry f xpt possily th outrf is tringl 2
lin sgmnt n this lin sgmnt nnot proprly intrst ny of th xisting gs of th grph. This itionl rstrition implis tht thr r vli g movs in th omintoril stting tht r no longr vli in th gomtri stting sin vnthough th grph rsulting ftr mov is plnr n simpl, it os not hv pln ming. S Figur 1. Lwson [15] show tht givn ny two nr-tringultions N 1 n N 2 m on th sm n points in th pln, thr lwys xists finit squn of g flips tht trnsforms th g st of N 1 to th g st of N 2. Hurto t l. [12] show tht O(n 2 ) flips r lwys suffiint n somtims nssry. Susquntly, Gltir t l. [9] show tht O(n) simultnous g flips r suffiint n somtims nssry. Not tht thr is isrpny twn th omintoril rsult n th gomtri on. In th omintoril stting, Wgnr [22] show tht vry tringultion on n vrtis n ttin from vry othr tringultion vi g flips. In th gomtri stting, Lwson [15] show tht only th nrtringultions tht r fin on th spifi point st n ttin vi g flips. For xmpl, in th point st shown in Figur 2, no plnr K 4 (omplt grph on 4 vrtis) n rwn on th givn point st without introuing rossing. In ft, in th gomtri stting, givn st of points in onvx position, th only pln grphs tht n rwn without rossing r outr-plnr. Givn point st Stright-lin ming of K 4 hs rossing Figur 2: Disrpny twn omintoril n gomtri stting. It is prisly this isrpny tht sprk our invstigtion. Th first qustion w sk is whthr or not thr xists simpl lol trnsformtion tht prmits th numrtion of ll n-vrtx tringultions in th gomtri stting. In orr to nswr this qustion, th lol trnsformtion must mor gnrl thn n g mov. Two ky ingrints n to spifi for this qustion. First, w n to spify th st of points P on whih ths grphs r m n on whih th trnsformtions n prform. To ovrom th isrpny with th omintoril stting, this st of points must hv th proprty tht vry n-vrtx tringultion hs stright-lin ming on n n-point sust of P. Suh st of points is ll univrsl point st. Shnyr [21] show tht th n n gri is univrsl point st for ll n-vrtx plnr grphs (s lso [8]). Thrfor, gri is nturl hoi for this stting. Howvr, using gri oms t ost sin thr r mny ollinr points in gri, w n to l spifilly with gnris. Dspit this ostl, w us gris s our univrsl point st n w outlin th xt gri sizs rquir for our rsults. All of our gri sizs r within onstnt of th optiml for stright-lin 3
mings of plnr grphs. It is importnt to kp th gri siz s smll s possil sin lrg gris hinr prtil pplitions of ths trnsformtions. Th son ingrint is th st of llowl lol trnsformtions. Bsis th g mov, th othr lol trnsformtion w us is point mov. A point mov is simply th moifition of th oorints of on vrtx. Th mov is vli provi tht ftr moving th vrtx to nw gri point, no g rossings r introu. f Eg mov: to f f Point mov vrtx. f Figur 3: Exmpl of n g mov follow y point mov 2 Trnsforming On Pln Tringultion to Anothr In this stion, w show tht O(n) point movs n O(n 2 ) g movs suffi to trnsform on pln n-vrtx tringultion into nothr. Th first tmpttion is to simply try to mimi omintoril flip in th gomtri stting with point n g movs. Howvr, this pproh prov to quit iffiult to hrtriz th point n g movs n to mimi on omintoril flip. In our proof, w rw is from Wgnr s originl rsult without mimiking omintoril flips. Essntilly, w show how to trnsform ny givn pln tringultion into nonil on, whih immitly implis th rsult. 4
Mor prisly, lt G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) with V 1 = V 2 = n two tringultions m in n n n gri. Lt th origin (0,0) of this gri th ottom lft ornr. Lt P 5n 5n gri with ottom lft ornr lot t ( 2n, 2n) n top right ornr lot t (3n, 3n). During th whol squn of movs, th lotion of vry point mov is gri point of P (i.. P is our univrsl point st). W show how to onstrut squn of O(n 2 ) g movs n O(n) point movs tht trnsforms oth G 1 n G 2 into nonil form. Th nonil tringultion is tringultion whr th outr f onsists of vrtis lot t ( 2n, 1),(3n, 1), n ( n/2, 3n). Th othr n 3 vrtis r lot t ( n/2,3n i),1 i n 3. Th two ottom ornr vrtis r strs jnt to ll othr vrtis n th grph inu y th rmining vrtis is pth whih w will ll th spin. Th nonil tringultion is shown in Figur 4. Not tht on w hv n ming of th nonil tringultion, th tul lotion of th oorints is no longr importnt us with O(n) point movs, it is firly simpl to mov from on ming of th nonil tringultion to ny othr s long s th spin is on vrtil gri lin n th outr-f forms tringl. ( n/2,3n) n n gri ( 2n, 1) (3n, 1) Figur 4: Illustrtion of th nonil tringultion n th initil gri. Bfor showing how to onstrut th squn of point n g movs, w n to stlish fw si uiling loks. On usful tool is th rsult y Hurto t l. [12] Thorm 1. [12] Lt T 1 n T 2 two tringultions whos vrtx st is st of n points in th pln. With O(n 2 ) g movs, T 1 n trnsform into T 2. Howvr, ky thnil lmm in thir proof of Thorm 1 is n ssntil tool in our work. 5
Lmm 1. [12] Lt T = (V,E) n ritrry nr-tringultion whos vrtx st is st of n points in th pln. Lt,, thr onsutiv vrtis on th outrf of T. Lt P th pth from to on th onvx hull of V \. With prisly k g movs, whr k is th numr of gs of T tht intrst P, w n trnsform T into tringultion tht ontins P. Not tht k is O(n). Nxt, w osrv two simpl fts out tringls tht will hlpful in th squl. Osrvtion 1. Lt () tringl in th pln. Lt x point ontin in th intrior of th tringl. Any lin through x whih hs oth n in on hlf-pln must hv in th othr n must intrst th lin sgmnts n. Osrvtion 2. Lt = (0,0), = (x 1,y 1 ) with x 1,y 1 > 0, = (x 1,y 1 + 1) n = (x 2,y 2 ) with x 2 > x 1 n y 2 (y 1 + 1)x 2 /x 1. Th point is ontin in th intrior of tringl (). W now sri squn of g movs n on point mov whih w will ll n px sli. Th stting for n px sli is th following. Lt,, th vrtis of 3-yl in tringultion G (i.. () is ithr f of G or sprting tringl in G). Lt x point suh tht x forms tringl with oth x n r on th sm si of hlf-pln fin y th lin through. Lt D ll th vrtis of G\ in (). Th st D is ontin in (x), n vry g in G intrst y sgmnt x or x is jnt to. S Figur 5 for n illustrtion. Figur 5: Apx Sli Lmm 2. With O(n) g movs n on point mov, vrtx n mov to point x. Proof. Lt D th vrtis of G\ in (). Lt C = 0, 1,..., k th lokwis orr of th onvx hull of D strting t = 0 n ning t k =. By Lmm 1, with O(n) g movs, w n onvrt th tringultion ontin in Th onvx hull of st of points is th smllst onvx polygon ontining th st. S ny stnr rfrn in Computtionl Gomtry [7] for n ovrviw. 6
to on whih ontins th sgmnts i n th gs of C. On this is omplish, Osrvtion 1 implis tht w n mov to x without introuing ny rossing sin C is ontin in oth () n (x) n y onstrution no othr g of G intrsts x or x. Thus, totl of O(n) g movs n 1 point mov suffi s rquir. To initit th whol pross, w n to show how givn tringultion m in th n n gri w n lwys mov th vrtis of its outrf to th oorints ( 2n, 1), (3n, 1) n ( n/2, 3n). In orr to ontinu, w n to fin th wg of onvx vrtx. Lt C onvx polygon on n vrtis. Lt v vrtx of C, n 1, 2 th two gs of C jnt to v. Givn th lin l 1 (rsp. l 2 ) fin y th npoints of 1 (rsp. 2 ), lt h 1 (rsp. h 2 ) th hlf-pln fin y this lin tht os not ontin th intrior of C. Th wg of v not W(v) is h 1 h 2. Th ngl of wg is th ngl twn th two rys fining th ounry of th wg. Th proprty of wg tht w will xploit throughout is th following: givn (), lt x point in W(). Not tht () (x). This ontinmnt proprty llows on to us px slis. Lmm 3. Givn tringultion G = (V,E) m in th n n gri, with O(n) g movs n t most 8 point movs, w n trnsform it into tringultion whos outrf hs oorints ( 2n, 1), (3n, 1) n ( n/2, 3n). All othr vrtis of G hv oorint vlus twn 0 n n (i.. thy r in th originl n n gri). Proof. Lt V = v 1,v 2,...,v n ll suh tht v 1, v 2 n v 3 r th vrtis on th outrf. Lt th oorints of h v i (x i,y i ) with 0 x i,y i n. Sin th outrf is tringl, th sum of th ngls of th wgs of its vrtis is π. This mns tht on of its vrtis prmits n px sli vrtilly (ithr up or own) n on of its vrtis prmits n px sli horizontlly (ithr lft or right). Not tht ths n th sm vrtx. Without loss of gnrlity, ssum tht v 1 n mov lft. All othr ss r symmtri. Th ft tht v 1 n mov lft mns tht th intrstion of horizontl lin through v 1 with W(v 1 ) is ry root t v 1 pointing lft. Mov v 1 to ( 2n,y 1 ). Aftr this mov, noti tht th ngl of th wg t v 1 is stritly lss thn π/2. This implis tht v 1 nnot mov vrtilly. Thrfor, on of v 2 or v 3 n mov vrtilly. Assum, without loss of gnrlity, tht v 2 n mov own. Agin, ll othr ss r symmtri. Mov v 2 to (x 2, 2n) h with n px sli. This mounts to O(n) g movs n 2 point movs. Nxt, pply two px slis to mov v 1 to ( 2n,n + 1) n v 2 to (n + 1, 2n). Both ths movs n ppli sin y onstrution no sgmnt with on npoint on th sgmnt [( 2n, 1),( 2n,n + 1)] n th othr npoint on sgmnt [( 1, 2n),(n + 1, 2n)] intrsts th originl n n gri. On ths movs hv n ppli, th outrf is suh tht v 3 llows n px sli up to (x 3,3n). On this hs n omplish, thr mor px slis pl th thr vrtis in nonil position. A totl of 8 px slis wr us giving th sir rsult. W now sri th min stp in th pross. Lt,, th vrtis of th outrf of tringultion G m on gri, suh tht n li on 7
th sm horizontl gri lin L 1, thr r t lst 5n 1 gri points twn n, th vrtx is ov n. Lt x point of th gri tht is not vrtx of G suh tht n x li on th sm vrtil gri lin L 2 suh tht th gri point z = L 1 L 2 is twn n with t lst 2n gri points twn n z n t lst 2n gri points twn n z. Th tringl () is th outrf of G n th point x stritly insi tringl (). All othr vrtis of G r stritly insi tringl (x). Thr r t lst n gri points on th sgmnt x. Noti tht if w r givn tringultion in n n n gri n pply Lmm 3, thn w mt th onitions spifi ov. Lmm 4. With O(n 2 ) g movs n O(n) point movs, w n trnsform G into nonil form. Proof. W pro y inution on th numr h of vrtis of G in (x). Bs Cs: h = 0. Th lmm hols trivilly sin no movs r rquir. Inutiv Hypothsis: 0 h k,k > 0. Assum tht 1 h 2 g movs n 2 h point movs suffi with onstnts 1 n 2. Inutiv Stp: h = k + 1. Lt r th first gri point low. Lt C = 0, 1,..., m+1 th lokwis orr of th onvx hull of th vrtis of G \ strting t = 0 n ning t = m+1. Apply Lmm 1, to onvrt G to tringultion ontining C n ll sgmnts i for vrtis i of th onvx hull. This is omplish with 3 k g movs for onstnt 3. Lt j j+1 th g of th onvx hull tht intrsts th vrtil lin through x. If th lin through x ontins vrtx of th onvx hull, ssum this vrtx is j. Thr r two ss to onsir. Cs: On of j or j+1 is vrtx of th onvx hull. Assum without loss of gnrlity tht j is onvx hull vrtx. Sin j is vrtx of th onvx hull, th points j 1, j, n j+1 r not ollinr. Sin th gri point r is in tringl ( j 1 j+1 ) y onstrution, w n pply n px mov to mov point j to r. Cs: Th g j j+1 is in th intrior of th onvx hull g = s t. Assum for th momnt tht th g hs positiv slop. Sin s is vrtx of th onvx hull, this mns tht s 1, s n s+1 r not ollinr. By Osrvtion 2, thr is gri point y on unit vrtilly ov s insi tringl ( s 1 s ). Apply n px mov to mov s to y. This rmovs th ollinrity from th onvx hull. Now th g y t is on th onvx hull. Romput th onvx hull n pply Lmm 1 so tht is jnt to ll gs of th onvx hull. Now w hv ru th sitution k to th prvious s. A symmtri rgumnt hols if hs ngtiv slop. Thrfor, with 4 k g movs n t most 2 point movs, w rmov on vrtx of G from (x), n mov it to r. Now, thr r only k vrtis of G rmining in th tringl (x). Apply Lmm 1 so tht r is jnt to ll vrtis on th onvx hull of G \ {,r}. W n now pply th inutiv hypothsis. Th totl numr of g movs is 1 k 2 + 4 k n th totl numr of point movs is 2 k+2. If w st 1 > 4 n 2 > 2, thn 1 k 2 + 4 k < 1 (k+1) 2 n 2 k + 2 < 2 (k + 1). Th lmm follows y inution. W r now in position to prov th min thorm of this stion. 8
Thorm 2. Givn n n-vrtx tringultion G = (V,E) m in th n n gri with stright-lin sgmnts s gs, with O(n 2 ) g movs n O(n) point movs (ll point movs sty within th gri [( 2n, 2n),(3n,3n)]), w n trnsform G into th nonil tringultion. Proof. Lt R rprsnt th points of th n n gri ontining G n lt P rprsnt th univrsl point st. First pply Lmm 3 to G. Thn, w n pply Lmm 4. Th thorm follows. An immit orollry is th following. Corollry 1. Givn two n-vrtx tringultions G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) m in th n n gri with stright-lin sgmnts s gs, with O(n 2 ) g movs n O(n) point movs (ll point movs sty within th gri [( 2n, 2n),(3n, 3n)]), w n trnsform G 1 into G 2 Rmrk: W not tht with littl r, our gri siz n ru to 3n 3n t th xpns of simpliity of xposition. W hos to kp th xplntions simpl in orr to sily onvy th min is rthr thn gt ogg own in tils. 3 Trnsforming On Tr to Anothr In this stion, w show tht O(n) point n g movs suffi to trnsform on tr into nothr n this is optiml s thr r pirs of trs tht rquir Ω(n) point n g movs to trnsform on into th othr. Lt G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) two trs m in th pln on n n n gri with V 1 = V 2 = n. Lt th origin (0,0) of this gri P th ottom lft ornr. During th whol squn of movs, th lotion of vry point mov is gri point of P. Th pproh is similr to tht us for tringultions, ut sin trs r simplr strutur, th numr of movs n th gri siz r ru. Avis n Fuku [1] show tht givn ny tr m in th pln, with t most n 2 g movs, this tr n trnsform into nonil tr. Th nonil tr thy us is th str from th lftmost vrtx. Morovr, th squn of g movs is suh tht h nw g os not intrst ny of th gs in th urrnt tr. Tht is, if T is th urrnt tr n th g mov onsists of ing T n lting f T, thn T is pln grph. W ll suh n g mov plnr g mov. Unfortuntly, w nnot us this rsult irtly, sin it is prov for points in gnrl position whr no thr points r ollinr n w r ling with trs m in th gri. Howvr, w moify thir rsult to ount for th ollinritis. Th nonil tr n no longr th str from th lftmost point sin thr my ollinritis. Lt p 1,p 2,...,p n th vrtx st of th givn tr T. Rll th points in th following mnnr. Lt p 1 th lftmost, ottommost point. Ll th othr points p 2,...,p n in sort orr ountr-lokwis roun p 1 so tht p 1 p 2 n p 1 p n r on th onvx hull, n if p 1,p i,p j r ollinr, thn i < j. Th nonil tr is th following: th g p 1 p i is in th tr if thr is no point p j,j i in th intrior of th sgmnt p 1 p i. If th sgmnt p 1 p i hs 9
points in its intrior, lt p k th intrior point losst to p i. Th sgmnt p k p i is in th tr. Not tht ssntilly this uils pths of ollinr vrtis from p 1. Th pths of ollinr vrtis shll rfrr to s tntls of p 1. S Figur 6 for n illustrtion. Figur 6: Cnonil Tr. Lmm 5. A tr T with n-vrtis m in th n n gri n trnsform into th nonil tr with n 2 g movs. Eh g mov is plnr. Proof. Lt T th givn tr m on th points p 1,...,p n ll s ov. Cll n g p i p j of T trnsvrsl g if th lin through p i p j os not ontin p 1. W pro y inution on th numr t of trnsvrsl gs. Bs Cs: t = 0. In this s, T is th nonil tr. Inutiv Hypothsis: t < k,k > 0. With t g movs, T n trnsform into th nonil tr. Inutiv Stp: t = k. Avis n Fuku [1] show tht for points in gnrl position, if T is not str from p 1, thr lwys xists n g p i p j T suh tht no othr g of T intrsts th intrior of th tringl (p 1 p i p j ). W nnot pply thir rsult irtly sin th points of T r not in gnrl position. Howvr, th sm rgumnt shows th xistn of trnsvrsl g p i p j suh tht for ny point p in th intrior of sgmnt p i p j, th sgmnt p 1 p os not intrst ny othr trnsvrsl g. Now, rmoving p i p j isonnts T into two omponnts C 1 ontining p i n C 2 ontining p j. Without loss of gnrlity, lt p 1 in C 1. Lt p 1 = x 1,x 2,...,x = p j th vrtis of T on th sgmnt p 1 p j. Sin p 1 C 1 n p j C 2, thr xists k suh tht x k C 1 n x k+1 C 2. A g x k x k+1 to th tr. Sin w hv ru th numr of trnsvrsl gs with on g mov, th rsult follows y inution. 10
Lmm 5 givs us th from to mov from ny on tr T 1 to ny othr tr T 2 fin on th sm point st with 2n 4 g movs sin w n trnsform on tr to th othr vi th nonil tr. Now, to trnsform tr m on on point st to nothr tr m on iffrnt point st, w n to prform som point movs. Givn n n-vrtx tr T m in th n n gri, w show how to trnsform it into pth m on vrtis (0,i), 0 i n 1. Lt p 1,p 2,...,p n th n points of T. Rll ths points so tht thy r sort y inrsing X oorint with p 1 ing th lftmost, ottommost point. If two points p i n p j r on th sm vrtil gri lin, thn i < j if p i is low p j. Now Lmm 5 implis tht T n trnsform to th pth p 1,p 2,...,p n with 2n 4 g movs. W ll suh pth monoton pth (s Figur 7 for n illustrtion of suh pth). Figur 7: A monoton pth. Lmm 6. A monoton pth m on th n n gri n trnsform to th nonil pth m on vrtis (0, i), 0 i n 1 with n point movs. Proof. By finition, th hlf-pln to th lft of th vrtil lin through th lftmost point is mpty. Thrfor, th lftmost, ottommost point n mov to ny gri point low it n to its lft. Mov it to (0,0). On this point is mov, th nxt lftmost, ottommost point n mov to (0,1). Th lmm follows y inution. Thorm 3. Givn two trs T 1 n T 2 m on th n n gri, with t most 4n 8 g movs n 2n point movs, T 1 n onvrt to T 2. All movs rmin in th originl gri. Proof. Th thorm follows from th isussion ov n Lmmt 6 n 5. In orr to show th lowr oun, tk n n-vrtx str n n n-vrtx pth h m on n iffrnt gri points. To onvrt th pth to str, w n t lst n 3 g movs sin ll vrtis of th pth hv gr t most 2 n th str hs vrtx of gr n 1. Similrly, sin non of th points of th str oini with th points of th pth, w n t lst n point movs to gt from th vrtx st of th pth to tht of th str. 11
Thorm 4. Thr xist pirs of trs T 1 n T 2 m on th n n gri tht rquir t lst n 3 g movs n t lst n point movs to trnsform on to th othr. 4 Trnsforming On Pln Grph to Anothr W now show how to gnrliz th rsults from Stion 2 to pln grphs. Givn two pln grphs G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) m in th n n gri with V 1 = V 2 = n n E 1 = E 2 = m, w show how to trnsform G 1 into G 2. W will ssum tht oth grphs r onnt. Th ovious pproh is to ummy gs to oth grphs until thy r tringultions. Thn, pply th prvious rsult n ignor th movs tht onrning ummy gs. Although this si pproh works, w n to rss fw tils long th wy. W will show how to trnsform G 1 into nonil form. Th prolm is tht sin G 1 is not tringultion, w n to spify prisly wht th nonil form is. Rll th nonil form for tringultions n ll its vrtis in th following wy. Lt p 1 n p 2 th lft n right ornrs of th outrf n lt p 3 th px. Ll th vrtis p 4,...,p n in sning orr on th spin from p 3. Ll th gs jnt to p 1 y 0,..., n 2 in lokwis orr roun p 1 with 0 = p 1 p 3 n n 2 = p 1 p 2. Ll ll th gs jnt to p 2 xpt g p 1 p 2 n p 2 p 3 y n 1,..., 2n 5, in ountr-lokwis orr with n 1 = p 2 p 4 n 2n 5 = p 2 p n. Now, th vlu of m trmins th shp of th nonil grph. Sin G 1 is onnt n plnr, n 1 m 3n 6. If m = n 1, thn th nonil grph is tr form y th pth from p 3 to p n long with th gs p 1 p 3 n p 2 p 3. If m > n 1, lt k = m n+1. Augmnt th nonil tr with th gs 1,..., k. Th first stp is to tringult G 1. Biolor th gs r n lu so tht th originl m gs r r n ll itionl gs r lu. Nxt, w show how to mov thr of th vrtis of G 1 to th oorints ( 2n, 1), (3n, 1) n ( n/2, 3n). Lmm 7. Givn nr-tringultion G m in th n n gri, with O(n) movs n t most 8 point movs, w n trnsform it into nr-tringultion whos outrf hs thr vrtis t oorints ( 2n, 1), (3n, 1) n ( n/2, 3n). All othr vrtis of G hv oorint vlus twn 0 n n (i.. thy r in th originl n n gri). Proof. Similr to th proof of Lmm 3 sin th outrf is onvx polygon so on of th vrtis of th polygon n mov vrtilly n on horizontlly. By pplying Lmm 7 to G 1, w onvrt it to to grph whos onvx hull onsists of 3 vrtis. Now, w n mor lu gs suh tht G 1 is tringultion n no longr nssrily nr-tringultion. This prmits us to pply Thorm 2, whih rsults in nonil tringultion with m r gs n th rmining r lu. Th r gs r not nssrily in nonil 12
position ut t most m g movs llow on to rshuffl th gs into orrt position. Thrfor, w hv th following: Thorm 5. Givn n n-vrtx pln grph G = (V,E) m in th n n gri with stright-lin sgmnts s gs, with O(n 2 ) g movs n O(n) point movs (ll point movs sty within th gri [( 2n, 2n),(3n,3n)]), w n trnsform G into th nonil pln grph. Corollry 2. Givn two n-vrtx pln grphs G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) m in th n n gri h hving m gs, with O(n 2 ) g movs n O(n) point movs (ll point movs sty within th gri [( 2n, 2n),(3n, 3n)]), w n trnsform G 1 into G 2 On spt of this pproh whih my unstisftory is tht throughout th squn, th grph my om isonnt vnthough w strt with onnt grph. This gs th qustion: is thr wy to gurnt tht in onvrting on pln grph into nothr, w rmin onnt throughout th whol squn? W nswr this in th ffirmtiv with slight inrs in th numr of g movs ut th totl numr of g movs rmins qurti. Not tht point movs o not hng th onntivity of grph. Thrfor, w solly n to onntrt on g movs. Th min tool w us for g movs in tringultions is Lmm 1. Th ky i is to mintin onnt spnning r grph ftr vry g flip. W prov th following. Lmm 8. Lt G n n-vrtx nr-tringultion. Lt sust of th gs of G olor r suh tht th grph inu y th r gs is onnt n spnning. Th rmining gs of G r olor lu. Lt n g of G to flipp. With t most 1 g mov, w n flip suh tht th grph inu y th r gs rmins onnt n spnning, ftr h of th g mov n g flip. Proof. Lt R th grph inu y th m r gs. W n to show tht w n flip n g of G suh tht R rmins onnt ftr th flip. Lt th g to flipp. If is lu, thn flipping os not fft th onntivity of th grph inu on th r gs. If is r, thn th only wy tht th onntivity of R is fft is if is ut g of R. Sin is in G, is jnt to t lst on tringulr f of G. Lt,, with = th thr vrtis fining this f. Th gs n nnot oth r sin this woul ontrit th ft tht is ut g. Sin is ut g, th ltion of from R isonnts th grph into two omponnts with n going to iffrnt omponnts. Without loss of gnrlity, ssum tht n r in iffrnt omponnts. Thn prforming n g mov in th r grph from = to, w hv nw st of m r gs tht form onnt n spnning sugrph of G. Essntilly, this mounts to oloring lu n r. Now, sin is lu, w n flip without ffting th onntivity of R. Thrfor, ftr on g mov, w n prform th flip. Th lmm follows. Sin Lmm 1 uss g flips in nr-tringultion, n ths r th only g movs w us, w onlu with th following. A ut g is n g whos ltion isonnts grph 13
Corollry 3. Givn two onnt n-vrtx pln grphs G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) m in th n n gri h hving m gs, with O(n 2 ) g movs n O(n) point movs (ll point movs sty within th gri [( 2n, 2n),(3n, 3n)]), w n trnsform G 1 into G 2 whil rmining onnt throughout th squn of movs. 5 Ll Trnsformtions In this stion, w rss th sm prolms in th ll stting, tht is w impos n initil mpping prior to th trnsformtion. Spifilly, givn two pln grphs G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ), w fin mpping φ : V 1 V 2. Now prform squn of g n point movs tht trnsforms G 1 into grph G 3 = (V 3,E 3 ) tht is isomorphi to G 2. Thr is mpping δ : V 1 V 3. In th unll s, w simply wnt G 3 to isomorphi to G 2. In th ll s, in ition, w wnt for vry vrtx x V 1, tht φ(x) = δ(x). 5.1 Ll Trs In th s of trs, th ll stting is firly strightforwr. W r givn two trs T 1 n T 2 s wll s mpping φ. Lmmt 5 n 6 imply tht oth T 1 n T 2 n onvrt into nonil tr, whih is pth on on row of th gri. Givn th two nonil pths, th only prolm is th lling. W show simpl wy to prmut th lls. Assum without loss of gnrlity tht th pth is on th X xis with vrtis t points (0,0),(2,0),(4,0),...,(2n 2,0). Mov (0,0) to (0,1). With n 1 g movs, onvrt th pth to str root t (0,1). Now, if w wish to prmut th lotion of th vrtx t (2i,0) with tht of (2j,0) simply mov th vrtx (2j,0) to (2i + 1,0), vrtx (2i,0) to (2j,0) n (2i + 1,0) to (2i,0). In this wy, w n sort th lls of th vrtis with O(n) g n point movs. Thorm 6. Givn two trs T 1 n T 2 m on th n n gri, n mpping φ of th vrtis of T 1 to th vrtis of T 2, with O(n) point n g movs, T 1 n onvrt to T 2 rspting th mpping. Proof. Follows from th isussion ov n Thorm 3. 5.2 Ll Tringultions n Pln Grphs In th ll stting for tringultions n pln grphs, w simply n to show tht givn ny lling of th vrtis of nonil tringultion, w mov th vrtis to gt iffrnt lling of th vrtis. Th i is similr to th prvious stion. W will show how to prmut th lls on th spin n how to mov ll from th spin to th outrf. First w show how to prmut th vrtis on th spin. In orr to o this, w simply n to show how to trnspos th position of two vrtis. W first moify th spin of th nonil tringultion suh tht twn vry vrtx thr is t lst on gri point. W lso mk sur tht th tringl of th outrf hs s ngls lss thn π/4. This will nsur tht thr is gri point in vry tringl y Osrvtion 2 n mk th following movs possil. 14
f f f Figur 8: Prmuting th Spin of th Cnonil Tringultion. Lt,,, four onsutiv vrtis on th spin n w wish to prmut n. S Figur 8. With on point mov, mov on gri point to th right. Now,,, form onvx quriltrl, so flip g for. Nxt, mov on gri point to th lft n flip g f for. Now mov to th sm hight s. This givs us th tringultion in th mil of Figur 8. Rvrs th stps to hng th orr of n on th spin. So with O(1) point movs n g movs, w n prmut th position of two onsutiv vrtis on th spin. Lmm 9. With O(n 2 ) point n g movs, w n go from ny prmuttion of th spin to ny othr prmuttion. Proof. Follows from th isussion ov n th ft tht O(n 2 ) trnspositions r suffiint to sort ny prmuttion of n numrs. Nxt, it is sy to s tht vrtx n mov from th spin to th outrf with on mov, s illustrt in Figur 9. Thn, with O(n) g movs n two point movs, th vrtx tht ws in th outrf, n mov to th ottom of th spin. f f f Figur 9: Moving vrtx from th spin to th outrf. This implis tht ll of th rsults on moving pln grphs trnslt to th ll stting t th itionl ost of O(n 2 ) point n g movs. Thorm 7. Givn two n-vrtx pln grphs G 1 = (V 1,E 1 ) n G 2 = (V 2,E 2 ) m in th n n gri h hving m gs, n mpping φ of th vrtis of G 1 to th vrtis of G 2, with O(n 2 ) g movs n O(n 2 ) point movs (ll point movs sty within th gri [( 2n, 2n),(3n,3n)]), w n trnsform G 1 into G 2 whil rspting th givn mpping n rmining onnt throughout th squn of movs. 15
6 Conlusion W hv shown how point n g movs suffi to trnsform ny pln grph to ny othr pln grph m in th pln with finit numr of movs on smll gri. As for furthr rsrh, it woul intrsting to ithr ru th numr movs us or fin mthing lowr ouns in th gnrl stting. Rfrns [1] D. AVIS AND K. FUKUDA, Rvrs srh for numrtion. Disrt Appli Mth., 65:21 46, 1996. [2] P. BOSE, J. CZYZOWICZ, Z. GAO, P. MORIN, AND D. R. WOOD, Prlll igonl flips in pln tringultions. Th. Rp. TR-2003-05, Shool of Computr Sin, Crlton Univrsity, Ottw, Cn, 2003. [3] R. BRUNET, A. NAKAMOTO, AND S. NEGAMI, Digonl flips of tringultions on los surfs prsrving spifi proprtis. J. Comin. Thory Sr. B, 68(2):295 309, 1996. [4] C. CORTÉS, C. GRIMA, A. MARQUEZ, AND A. NAKAMOTO, Digonl flips in outr-tringultions on los surfs. Disrt Mth., 254(1-3):63 74, 2002. [5] C. CORTÉS AND A. NAKAMOTO, Digonl flips in outr-klin-ottl tringultions. Disrt Mth., 222(1-3):41 50, 2000. [6] C. CORTÉS AND A. NAKAMOTO, Digonl flips in outr-torus tringultions. Disrt Mth., 216(1-3):71 83, 2000. [7] M. DE BERG, M. VAN KREVELD, M. OVERMARS, AND O. SCHWARZKOPF, Computtionl Gomtry: Algorithms n Applitions. Springr-Vrlg, Brlin, Grmny, 2n n., 2000. [8] H. DE FRAYSSEIX, J. PACH, AND R. POLLACK, How to rw plnr grph on gri. Comintori, 10(1):41 51, 1990. [9] J. GALTIER, F. HURTADO, M. NOY, S. PÉRENNES, AND J. URRUTIA, Simultnous g flipping in tringultions. Intrnt. J. Comput. Gom. Appl., 13(2):113 133, 2003. [10] Z. GAO, J. URRUTIA, AND J. WANG, Digonl flips in lll plnr tringultions. Grphs Comin., 17(4):647 657, 2001. [11] F. HURTADO AND M. NOY, Grph of tringultions of onvx polygon n tr of tringultions. Comput. Gom., 13(3):179 188, 1999. [12] F. HURTADO, M. NOY, AND J. URRUTIA, Flipping gs in tringultions. Disrt Comput. Gom., 22(3):333 346, 1999. [13] H. KOMURO, Th igonl flips of tringultions on th sphr. Yokohm Mth. J., 44(2):115 122, 1997. 16
[14] H. KOMURO, A. NAKAMOTO, AND S. NEGAMI, Digonl flips in tringultions on los surfs with minimum gr t lst 4. J. Comin. Thory Sr. B, 76(1):68 92, 1999. [15] C. LAWSON, Softwr for 1 surf intrpoltion. In J. RICE,., Mthmtil Softwr III, pp. 161 194, Ami Prss, Nw York, 1977. [16] A. NAKAMOTO AND S. NEGAMI, Digonl flips in grphs on los surfs with spifi f siz istriutions. Yokohm Mth. J., 49(2):171 180, 2002. [17] S. NEGAMI, Digonl flips in tringultions of surfs. Disrt Mth., 135(1-3):225 232, 1994. [18] S. NEGAMI, Digonl flips in tringultions on los surfs, stimting uppr ouns. Yokohm Mth. J., 45(2):113 124, 1998. [19] S. NEGAMI, Digonl flips of tringultions on surfs, survy. Yokohm Mth. J., 47:1 40, 1999. [20] S. NEGAMI AND A. NAKAMOTO, Digonl trnsformtions of grphs on los surfs. Si. Rp. Yokohm Nt. Univ. St. I Mth. Phys. Chm., (40):71 97, 1993. [21] W. SCHNYDER, Eming plnr grphs on th gri. In Pro. 1st ACM- SIAM Symp. on Disrt Algorithms, pp. 138 148, 1990. [22] K. WAGNER, Bmrkung zum Virfrnprolm. Jr. Dutsh. Mth.- Vrin., 46:26 32, 1936. [23] T. WATANABE AND S. NEGAMI, Digonl flips in psuo-tringultions on los surfs without loops. Yokohm Mth. J., 47:213 223, 1999. 17