A 43k Krnl for Plnr Dominting St using Computr-Ai Rution Rul Disovry John Torås Hlsth Dprtmnt of Informtis Univrsity of Brgn A thsis sumitt for th gr of Mstr of Sin Suprvisor: Dnil Lokshtnov Frury 2016
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I think vryon shoul lrn how to progrm omputr us it ths you how to think - Stv Jos
Aknowlgmnts First n formost I must thnk my suprvisor, Dnil. Your knowlg n guin hs n invlul uring th work with this thsis. You hv lwys n vill for qustions; wkns, timzons n ontinnts hv nvr hinr m from rhing out. You r grt thr, n hv th ility to s simpl n lgnt solutions whr I s prolms; tht is somthing I rlly mir. I must lso thnk th stunts n fulty t th Dprtmnt of Informtis, Univrsity of Brgn. You hv rt n nvironmnt of xitmnt n uriosity out th worl of Computr Sin, whih hs l m to lov ing prt of th fil. A spil thnks to th Algorithms group, whos fr pizz got m intrst in omptitiv progrmming n lgorithms in th first pl. Thnks to Simn, Mgnr n Erik; you hv n grt tm mts n mntors uring progrmming omptitions. It hs n fun n xtrmly vlul xprins. Of ours my fmily must mntion. My got m intrst in omputrs in th first pl, y ringing omputr mgzins n ol omputrs to tmpr n ply with, hom from work. My prnts hv lwys lt m o wht I lov, ut rmin m to gt som slp whn tht is u. An Christin, my girlfrin, thnk you for kping up with m in th lst wk for th lin. Now I m finlly oming hom.
Contnts I Introution n prliminris 1 1 Introution 3 1.1 Bkgroun n Thsis Ovrviw............................. 4 1.2 Trminology n Prliminris.............................. 5 1.2.1 Mthmtil Nottion.............................. 6 1.2.2 Points in th Pln................................. 6 1.2.3 Grphs....................................... 8 1.2.4 Pln Grphs.................................... 10 1.2.5 Dision Prolms................................. 12 1.2.6 Algorithms n Runtim............................. 12 1.2.7 Dominting St................................... 13 1.2.8 Effiint Algorithms n NP-hrnss...................... 13 1.2.9 Coping with NP-hrnss............................. 13 2 Fix Prmtr Trtility n Krnliztion 15 2.1 Fix Prmtr Trtl Algorithm for Vrtx Covr............... 15 2.2 Krnls........................................... 17 2.2.1 Rution Ruls.................................. 18 II A Linr Krnl for Plnr Dominting St 19 3 Plnr Dominting St 21 3.1 Nighorhoo of Vrtx................................. 21 3.2 Rution Ruls...................................... 22 3.3 Rgions n Rgion Domposition s Trt y Alr t l.............. 23 4 Domposing th Grph 27 4.1 Egs in Pln Emings................................ 27 4.2 Wlks in Pln Emings................................ 28 4.3 Rgions........................................... 32 4.4 Grph Enrihmnt..................................... 33 4.5 Rgion Domposition................................... 33 4.5.1 Mximl Rgion Domposition.......................... 35 5 A Linr Krnl 39 5.1 Th Krnl of Alr t l.................................. 41 6 A Smllr Krnl 43 i
6.1 Vrtx Flipping....................................... 43 6.2 Uppr Boun Outsi Rgions.............................. 44 6.3 Bouning Rgion Siz................................... 45 6.4 An Improv Krnl Uppr Boun............................ 46 III Computr-Ai Rution of Rgions 49 7 A 43k krnl for Plnr Dominting St 53 7.1 Dfinitions.......................................... 53 7.2 A 43k Krnl........................................ 57 8 Computr-Ai Rution of Dth Rgions 59 8.1 Mor Dfinitions...................................... 59 8.2 Rution Ruls Insi Rgion............................. 61 9 Boun Insi Innr n Singl Rgions 63 9.1 Innr rgions........................................ 63 9.1.1 Bouning th Siz of n Innr Qusi-Rgion................... 65 9.2 Singl Qusi-Rgions.................................... 67 9.3 Outr Rgions Without Possil Domintors...................... 68 9.4 Bounry Sizs n Vrtis 2............................... 69 10 Splitting Rgions Into Smllr Prts 71 10.1 Fully Enumrt Rprsnttiv Sts.......................... 71 10.2 Orr of Enumrtion................................... 74 11 Enumrtion of Fully Enumrt Rprsnttiv Sts 75 11.1 Innr rgions........................................ 75 11.2 Enumrting Singl Rgions................................ 75 11.2.1 (2, 1) s -rgions.................................... 75 11.2.2 (2, 2) s -rgions.................................... 76 11.2.2.1 (2, 2) s (1, )(,2)-rgions........................... 76 11.3 (3, 2) s -rgions........................................ 77 11.3.1 (3, 2) s (1,2,3)(1, )-rgions............................... 77 11.3.2 (3, 3) s (1,2,3)(1,2,3)-rgions.............................. 78 11.4 (2, 1) n -rgions........................................ 79 11.5 (2, 2) s,n -rgions....................................... 79 11.6 (2, 1)-rgions........................................ 79 11.7 (2, 2)-rgions........................................ 80 11.8 (3, 1)-rgions........................................ 81 11.9 (3, 2)-rgions........................................ 82 11.10(3, 3)-rgions........................................ 84 12 Implmnttion n Rsults 89 13 Conlusions 91 13.1 Opn Prolms....................................... 91 Biliogrphy 92 ii
Prt I Introution n prliminris 1
Chptr 1 Introution Imgin this snrio: You r th monrh in ountry with svrl smll towns sttr throughout th lns. Th nighoring towns r onnt y ros, whih n us to riv ffiintly twn thm. Your popl think you r n outstning monrh, n you wnt to kp it tht wy, us it just fls so goo ing worshipp, s shown in Figur 1.1. Evrything sms to going smoothly, ut sunly th towns strt hving prolms with osionl firs, n for som rson th houss urn to th groun quit fst. You i to pl fir sttions in som of th towns, suh tht vry town ithr hs its own fir sttion, or nighoring town with on. In tht wy Figur 1.1: Fls so goo to king. you mk sur tht in s of fir, thr s fir truk ry in th town, or svior n om from n jnt town. Howvr, uiling fir sttions is xpnsiv, n you n to sv goo hunk of your txpyrs mony so you n uil nw stl, in s you shoul n on. Wht is th minimum numr of fir sttions you n uil to gt ri of your prolm? An xmpl of n instn of this prolm is shown on th lft in Figur 1.2. How hr n this? You quikly rliz tht thr r no ovious positions to pl your fir sttions, so you sort of n to try iffrnt plmnts, n onvin yourslf tht your plmnt is optiml. On th right solution is shown, proving tht you n solv th prolm using only 3 fir sttions. In this prtiulr s you n prov with rltiv s tht no smllr solution is possil, ut this is not lwys tht sy. 3
Figur 1.2: A grph n its miniml Dominting St olor in grn. A ulltproof wy to fin th optiml solution is to try ll possil plmnts of fir sttions, n pik th smllst solution foun. Th prolm with this pproh is tht th numr of possil plmnts w hv to try oms stronomilly ig if th numr of towns is lrg. This is wht is known s rut-for pproh, n it is gnrlly onsir vry for ll othr thn smll prolm instns. This prtiulr prolm is known s th Dominting St prolm, n hs n hvily stui, oth for its mny prtil pplitions n its thortil spts. Th Dominting St prolm is importnt in mny inustris; tlphon oprtor pling rio towrs t rnom hoping to hiv goo ovrg woul quikly run out of mony; in soil ntwork thory solving th prolm n l to insight in popls influn on h othr[26]; n mny prolms in logistis, istriution n trnsporttion n mol in this wy. Dspit th grt fforts of mny popl sin th Dominting St prolm ws first formliz in th 1950 s[15] no ffiint lgorithm for solving it hs n foun. Th rson for this ws foun in th 70 s, whn svrl rsrhrs pulish pprs on th thory on NP-ompltnss[7, 20, 18]. It turns out tht Dominting St is mong ths NP-omplt prolms, n it is liv tht no ffiint lgorithms xist to solv thm. Evn though w liv tht it is not possil to solv th prolm ffiintly in th gnrl s, mny rl-lif instns of th prolms hv strutur tht n mk th prolm mor trtl. Looking k t th fir sttion xmpl, w n osrv tht th instn in this s is mp showing th towns rwn in th pln, with no ros rossing h othr. This informtion n us to improv on our lgorithms whn trying to fin solution. As w shll s, th plnr vrsion of th prolm, Plnr Dominting St, is onsir sir thn th gnrl prolm. In som ss w n isovr prts of th input instn tht r sir to solv thn th rst, n w n rmov ths prts ffiintly from th instn. W will show tht for lrg instn to th Plnr Dominting St prolm w n lwys ru th instn to siz just pning on th siz of th optiml Dominting St. This rmining prt w r lft with is ll th krnl of th instn. 1.1 Bkgroun n Thsis Ovrviw In this stion, w will provi rif ovrviw of th thsis, inluing som kgroun n rsults. This ovrviw mks us of fw si onpts from Prmtriz Complxity. Ths onpts r xplin in th prliminris prt. 4
In 2004, Alr, Fllows, n Nirmir[2] prsnt krnliztion lgorithm for Plnr Dominting St, n m up with th onpt of Rgion Domposition for plnr grphs. This is wy of iviing plnr grphs into rgions, n svrl pprs following thirs hv m us of this prtiulr thniqu for showing krnls for prolms on plnr grphs [1, 6, 11, 12, 13, 21, 23]. Unfortuntly thr r som miguitis in th finitions of th or onpts of th Rgion Domposition of Alr t l., n w o not s wy to rsolv ths miguitis suh tht th proof of th lmm s pulish in [2, Lmm 6] is orrt. For this rson, thr hs n som unrtinty [17] s to whthr th sttmnts m using Rgion Domposition n its thorms r tru, or whthr thy, n th rsults uiling on thm, shoul ronsir. Th first min ontriution of this thsis is to show tht minor moifitions to th finitions, thorm sttmnts n proofs of Alr t l. is nough to mk thir thorms n rsults go through. Our moifitions r onsistnt with th us of th Rgion Domposition thniqu in othr work w r wr of, giving ths rsults soli fountion. Our moifi finition of Rgion Domposition togthr with th proofs of Alr t l. show tht if plnr grph hs ominting st of siz k, thn thr xists Rgion Domposition of th grph hving t most 3k rgions. Togthr with th ouns of t most 55 vrtis insi h rgion n 170k vrtis in totl outsi rgions from Alr t l., this rsults in krnl on 335k vrtis. On of th pprs tht s thir proofs on th Rgion Domposition thniqu of Alr t l., ws th 2007 ppr y Chn, Frnu, Knj, n Xi[6]. Thy introu som itionl rution ruls n y n improv nlysis uppr oun th numr of vrtis in vry rgion y 16 n th vrtis outsi rgions to 19k. This yils 67k krnl for Plnr Dominting St. Our son min ontriution is to improv this rsult y introuing on xtr rution rul n y moifying th ming of th grph. Doing this w r l to oun th numr vrtis outsi rgions to 7k in totl, rsulting in 55k krnl. Chn t l. wr l to oun th siz of th rgions to 16 y using svrl rution ruls n n xtnsiv s nlysis. W utomt this s nlysis pross y th us of omputr progrm tht signs rution ruls y xhustivly srhing for ru rgions, n provs th orrtnss of th gnrt rution ruls s it runs. In th pross it kps trk of th iggst non-ruil rgions foun, rriving t th rsult tht ny rgion n ru to n quivlnt rgion of siz 12 or lss. This givs ris to 43k krnl. B sur to r th Wors of Wrning t th strt of Prt III, s w finish th oing of th omputr progrm it too los to th sumission t. Not tht th srh for ru rgions oul hv n on y nïvly numrting ll rgions up to th siz of 16, n kping th ru ons. But sin this will tk yrs, if not s, vn on supr-omputr, w o n xtnsiv nlysis to sp up th pross, mking it possil to rriv t th rsult in fw ys on smi-powrful prlll omputr. Chptrs 1-2 introu th Dominting St prolm n nssry nottion. In Chptrs 3-5 w prov our Rgion Domposition thorms n rriv t th 335k krnl. In Chptr 6 w improv this to 55k. Finlly, in Chptr 7 n out w sri our pross for oing omputr-i rution of rgions, furthr improving th krnl to 43k. 1.2 Trminology n Prliminris In this stion w will introu th rr to most of th or onpts w r touhing on throughout th thsis. Mny of th onpts n foun sri in ny goo introutory ooks to Disrt Mthmtis n Algorithms [25, 9] so w will only rifly xplin th most si ons, n spn most of our tim sriing thos mor prtiulrly rlvnt for our isussion of th Plnr Dominting St prolm. 5
1.2.1 Mthmtil Nottion W will frly mk us of stnr st nottion, s foun in Rosn[25]. For th sk of onvnin w will rstt th most ommon ons hr, fin tilor to our us. Dfinition 1.1 (Multist). A multist is n unorr olltion of lmnts. Dfinition 1.2 (St). A st is multist whr h lmnt pprs only on. Dfinition 1.3 (Univrs). A univrs is th st tht ontins ll th lmnts unr onsirtion in givn sitution. As n xmpl, if w r looking t oprtions on sts of positiv intgrs, th univrs in this sitution will prisly th st of ll positiv intgrs. Lt A, B two sts from som univrs. W hv th following finitions Mmr of st If th lmnt x is in th st A, w sy tht x is mmr of A, n not this y x A. A non-mmr y is not y A. Crinlity Th numr of mmrs of A is ll th rinlity of A, not A. Empty st A = if A = 0. Sust A B, if for vry x A, w hv x B. Equlity A = B, if A B n B A. Propr sust A B if A B n A B. Union A B = {x x A x B} Intrstion A B = {x x A x B} W will n th finitions for infimum n suprmum on th st of rl numrs: Dfinition 1.4 (Infimum). Lt S R. Th infimum of S, not inf(s), is grtst lmnt in R tht is not grtr thn ny lmnt in S, if suh n lmnt xists. For instn, lt S = (0, 10]. Thn inf(s) = 0. Th finition of suprmum is similr, ut this on w will only n for th nturl numrs: Dfinition 1.5 (Suprmum). Lt S N. Th suprmum of S, not sup(s), is smllst lmnt in N tht is not smllr thn ny lmnt in S, if suh n lmnt xists. 1.2.2 Points in th Pln To l to tlk out plnrity of grphs ltr, w will n nottion rgring points in th pln. Dfinition 1.6 (Crtsin oorint systm R 2 ). W sy tht R 2 is th st of ll pirs (x, y) whr x, y R. R 2 is ommonly rfrr to s th pln. Th mmrs of R 2 r ll points. Not tht w oftn will not st of points simply s point st. Dfinition 1.7 (Point-to-point istn). Lt p 1 = (x 1, y 1 ) n p 2 = (x 2, y 2 ) two points from R 2. Th Eulin istn, or simply istn, twn th two points is ist(p 1, p 2 ) = (x1 x 2 ) 2 + (y 1 y 2 ) 2. Dfinition 1.8 (Point st-to-point st istn). Lt A, B R 2 two point sts. Th istn twn th sts is ist(a, B) = inf{ist(, ) A, B}. W sy tht A n B r sprt y th istn ist(a, B). 6
Th rson w r using infimum for this finition is so w n sy tht two point sts tht r infinitly los in R 2, ut not intrsting, r sprt y istn of 0. Figur 1.3: Th istn twn two point sts is th smllst istn twn ny two points, on from h st. Whn oing ition n slr multiplition on points, w trt thm s rgulr vtors, n prform th oprtions s fin in Ly[19]. Dfinition 1.9 (Lin sgmnt). Lt p 1, p 2 two points in R 2. Th linsgmnt twn p 1 n p 2 is th st of points L(p 1, p 2 ) = {λp 1 + (1 λ)p 2 0 λ 1}. p 1 n p 2 r ll th npoints of th linsgmnt. p 2 p 1 Figur 1.4: A linsgmnt is th st of ll points on th lin twn th two npoints p 1 n p 2. Dfinition 1.10 (Simpl polygon lin). A simpl polygon lin P is point st P = L(p 1, p 2 ) L(p 2, p 3 ),..., L(p t 1, p t ) suh tht th following proprtis r stisfi: p 1, p 2,..., p t is finit squn of points whr p i p j for ll i j for 1 < i < j < t w hv L(p i 1, p i ) L(p j, p j+1 ) = for 1 < i < t w hv L(p i 1, p i ) L(p i, p i+1 ) = {p i } p 1 n p t r ll th npoints of P, n w writ P = P (p 1, p t ). f Figur 1.5: Th simpl polygon lin is m up of lin sgmnts twn th points through f. n f r th npoints of th simpl polygon lin. 7
Dfinition 1.11 (Connt points). W sy tht pir of points p 1, p 2 is onnt in point st S if thr xists simpl polygon lin P S suh tht p 1, p 2 P. Dfinition 1.12 (Connt pointst). A point st S is onnt if vry pir p 1, p 2 S is onnt in S. Dfinition 1.13 (Simpl polygon). A simpl polygon P is st of points s.t. thr xists simpl polygon lin P with npoints p 1, p t, n linsgmnt L(p 1, p t ) s.t. L(p 1, p t ) P = {p 1, p t }, n w hv P = P L(p 1, p t ) p 2 p 3 p 5 p 1 p 6 p 4 Figur 1.6: A simpl polygon. For onvnin, whn w sy just polygon lin or polygon, w will rfr to th finitions ov, mning simpl polygon lin n simpl polygon. Dfinition 1.14 (F). Lt P simpl polygon. Th inlusion-wis mximl onnt point sts of R 2 \ P r ll fs. Th infinit on is ll th xtrior f of P, whil th finit on is ll th intrior f of th polygon. Dfinition 1.15 (Unit siz isk). A unit siz isk with ntr p R 2 is th st of points 1.2.3 Grphs {p R 2 ist(p, p ) 1} Dfinition 1.16 (Vrtx). A vrtx v is singl lmnt from som univrs of lmnts U. Dfinition 1.17 (Eg). An g is n unorr pir of two vrtis u, v, writtn = (u, v). W sy tht is inint to u n v, lso tht gos twn u n v. u n v r th npoints of. Dfinition 1.18 (Simpl grph). A simpl grph G is pir V, E, oftn writtn G = (V, E). V is st of vrtis, n is ll th vrtis of G. E is st of gs twn vrtis in V, n is ll th gs of G. Mor spifilly, E {(u, v) u, v V, u v}. V n E r oftn writtn V (G) n E(G) rsptivly, whn it is miguous whih grph thy long to. Dfinition 1.19 (Multigrph). W sy grph G = (V, E) is multigrph if w llow E to multist. As w will mostly work with multigrphs from this point forwrs, w might somtims just writ grph, mning multigrph. Not tht our finition of multigrph os not llow slf-loops (i.. n g (u, v) s.t. u = v). Dfinition 1.20 (Ajnt vrtis). Lt G = (V, E) grph, n lt u, v V two vrtis in G. W sy tht u n v r jnt if thr is n g in G inint to thm oth. Tht is, if (u, v) E. 8
Dfinition 1.21 (Nighorhoo of vrtx). Lt G = (V, E) grph, n lt v V vrtx in G. W not th nighorhoo of v s N(v), n it onsists of ll th vrtis jnt to v in G. Th los nighorhoo of v is not y N[v] = {v} N(v). Dfinition 1.22 (Nighorhoo of st of vrtis). Lt G = (V, E) grph, n lt S V st of vrtis in G. As for singl vrtx, w not th los nighorhoo of S s N[S] = N[s] n th nighorhoo of S s N(S) = N[S] \ S. s S Dfinition 1.23 (Dgr of vrtx). Lt G = (V, E) grph, n lt v V vrtx in this grph. Th gr of v is th numr of gs from E inint to v, n w not it y g(v). Not tht sin w in som ss will llow multipl gs, th gr of vrtx is not lwys th sm s th numr of nighors it hs, s is th s for simpl grphs. Dfinition 1.24 (Inu sugrph). Lt G = (V, E) grph, n lt S V st of vrtis in G. Th sugrph inu y S is th grph hving vrtis V = S n gs E = {(u, v) E u, v S}, n is not G[S] = (V, E ). Dfinition 1.25 (Dlting vrtis of grph). Lt G = (V, E) grph, n lt S V. Th grph G S is th grph G = (V, E ), whr V = V \ S n E = {(v 1, v 2 ) E v 1 S v 2 S}. Dfinition 1.26 (Disjoint union of grphs). Lt G 1 = (V 1, E 2 ), G 2 = (V 2, E 2 ) two grphs suh tht V 1 V 2 =. Th isjoint union of G 1 n G 2, not G 1 + G 2, is th grph G = (V 1 V 2, E 1 E 2 ). Dfinition 1.27 (Aing gs to grph). Lt G = (V, E) grph n lt E 1 {(u, v) u, v V, u v} st of gs. Th grph G otin y ing th gs E 1 to G is th grph G = G + E 1 = (V, E E 1 ). Dfinition 1.28 (Suprgrph). Lt G = (V, E) grph. A grph G = (V, E ) is suprgrph of G if V = V n E E. Dfinition 1.29 (Wlk). Lt G = (V, E) grph. A wlk W of lngth k in G is squn (v 1, 1, v 2, 2,..., v k, k, v k+1 ) whr v i V, j E, suh tht j {1,..., k}, j = (v j, v j+1 ). Dfinition 1.30 (Simpl wlk). A simpl wlk is wlk whr ll th vrtis in th squn r istint. This is ommonly know s pth. Dfinition 1.31 (Cyli wlk). A yli wlk C is wlk W = (v 1, 1,..., v k, k, v k+1 ) suh tht v 1 = v k+1. Dfinition 1.32 (Cliqu). Lt G = (V, E) grph, n lt C V st of vrtis in G. W sy tht C is liqu if ll vrtis in C r jnt, tht is u C, v C, (u, v) E. A liqu on n vrtis is not K n. Dfinition 1.33 (Inpnnt st). Lt G = (V, E) grph, n lt I V st of vrtis in G. W sy tht I is n inpnnt st if no vrtx in I is jnt to ny othr vrtx in I, i.. u I, v I, (u, v) E. Dfinition 1.34 (Prtition of grph). Lt G = (V, E) grph, n lt A V n B = V \ A two isjoint susts of th vrtis of G. W sy tht (A, B) is prtition of th grph. Dfinition 1.35 (Biprtit grph). Lt G = (V, E) grph. W sy tht G is iprtit if thr xists prtition (A, B) of V (G) suh tht A is n inpnnt st, n B is n inpnnt st in G. (A, B) is ll iprtit prtition of G. Dfinition 1.36 (Complt iprtit grph). Lt G = (V, E) iprtit grph with iprtit prtition (A, B). W sy tht G is omplt iprtit grph if vry vrtx in A is jnt to vry vrtx in B, i.. A, B, (, ) E. A omplt iprtit grph hving prtition (A, B) whr A = n n B = m is not K n,m. 9
ɛ Figur 1.7: A plnr grph n lwys rwn with gs sprt y istn ϵ > 0. 1.2.4 Pln Grphs Th st of plnr grphs is th typ of grphs w ll turn most of our fous to in this thsis. W will now stt som finitions rgring plnr grphs tht will usful for th rst of th isussion. Dfinition 1.37 (Pln ming). A pln ming of grph G = (V, E) is four-tupl E = (U E, P E, f E, g E ), whr U E is st of unit siz isks, P E is st of polygon lins, f E : V U E is mp tht mps ny vrtx of V to unit siz isk in th pln, n g E : E P E is mp tht mps ny g in E to polygon lin in th pln, suh tht thr xists n ϵ > 0 n th following onitions r stisfi: 1. For vry pir of istint vrtis v 1, v 2 V thir unit isks r sprt y ϵ, i.. ist(f E (v 1 ), f E (v 2 )) ϵ 2. For vry pir of istint gs 1, 2 E, thir polygon lins r sprt y ϵ, i.. ist(g E ( 1 ), g E ( 2 )) ϵ 3. For vry g E n for vry vrtx v V not inint to, th unit isk of v n th polygon lin of is sprt y ϵ, i.. ist(f E (v), g E ()) ϵ 4. For vry g E n for vry vrtx v V inint to, th unit isk of v n th th polygon lin of will touh xtly in th point p, whr p is on of th npoints of th polygon lin of, i.. f E (v) g E () = {p} In simpl trms th first onition sys tht no two vrtis shoul rwn intrsting in th pln, th son tht no gs shoul ross in th rwing, n th lst two tht n g shoul touh only th unit isks of its npoints in th rwing. Not tht th finition of pln ming is tilor to our us ltr in th thsis, n might look it iffrnt from th stnr wy of fining it. Vry oftn in th litrtur, vrtis mps to points in th pln, n gs to stright lins or urvs. It is howvr possil to show tht ths finitions r quivlnt, ut w ll omit th proofs, n rfr to Mohr n Thomssn[22] for th stnr wy of fining grphs on surfs. Dfinition 1.38 (Plnr grph). A grph is plnr if it hs pln ming. Dfinition 1.39 (Pln grph). A pln grph G is grph G togthr with pln ming E of G: G = (G, E). 10
Dfinition 1.40 (Drwing of pln grph). Lt G = (V, E) plnr grph, n E = (U E, P E, f E, g E ) n ming of G. Th st of points E(G) = ( f E (v)) ( g E ()) is ll th rwing of G. v V Dfinition 1.41 (Pln suprgrph). Lt G = (G, E) pln grph. A pln suprgrph of G is suprgrph G of G n n ming E of G suh tht for ll u V (G), f E (u) = f E (u) n for ll E(G), g E () = g E (). Dfinition 1.42 (F in pln grph). Th fs of pln grph (G, E) r th mximl onnt sts of points of R 2 \ E(G). Th infinit f of pln grph is ll th xtrnl f or outr f. Dfinition 1.43 (Vrtx inint to f). Lt G grph with pln ming E = (U E, P E, f E, g E ), n lt f f in this grph. A vrtx v E is inint to th f f if ist(f, f E (v)) = 0. Dfinition 1.44 (Eg inint to f). Lt G grph with pln ming E = (U E, P E, f E, g E ), n lt f f in this grph. An g E is inint to th f f if ist(f, g E ()) = 0. Osrv tht n g n inint to t most two fs. Dfinition 1.45 (Siz of f). Th siz of f f in pln grph is f = w f (), whr E 0, if not inint to f w f () = 1, if is inint to f n som f othr thn f 2, othrwis Osrv tht h g in th grph will ontriut 2 to th sum of th siz of ll fs in th grph, n w hv tht f = 2 E f F whr F is th st of fs in th ming. A wll-known thorm tht will usful is th following. Thorm 1.46 (Eulr s formul [3, Thm 3.7]). Lt (G = (V, E), E) onnt pln grph, n lt F th st of fs in th ming. W hv tht V + F E = 2 E Using th ov thorm w n riv nothr usful on: Thorm 1.47 (Biprtit Plnr Grph Lmm). Lt G simpl plnr grph with iprtition (A, B) whr B, g() 3. Thn B 2 A. Proof. Lt n = V (G) = A + B th numr of vrtis, lt m = E 3 B th numr of gs, n lt f = F th numr of fs in G. Sin th grph is simpl n iprtit, f in th grph must hv siz of t lst 4, mning 4 f 2m or f m 2. Using this togthr with Eulr s formul yils: m = n + f 2 n + m 2 2 or m 2n. From for w hv 3 B m 2n = 2 A + 2 B = B 2 A. Th fmous thorm of Kurtowski sys tht grph is plnr if n only if it osn t ontin suivision of th liqu on 5 vrtis, K 5, or th omplt iprtit grph K 3,3 s sugrph. W won t fin suivisions hr, n rfr to Anrson[3] for this. Th importnt ft is tht this mks plnrity hking rltivly sy, n numr of ffiint lgorithms for hking plnrity 11
xist[16, 5]. This will usful for us whn w ltr wnt to gnrt mny plnr grphs on omputr, s w ffiintly n hk whthr th gnrt grphs r plnr. A thorm w will us ltr is on vry similr to Kurtowski s, rgring outrplnrity: Dfinition 1.48 (Outrplnr grph). A grph is outrplnr if it is hs pln ming whr vry vrtx is inint to th xtrnl f. Thorm 1.49. A grph G is outrplnr if n only if it hs no suivision of K 4 or K 2,3 s sugrph. W rfr to Distl[10] for th proof. 1.2.5 Dision Prolms Dfinition 1.50 (Dision Prolm). A ision prolm L is sust L Σ, whr Σ is fix siz lpht, n Σ nots th th st of ll finit strings ovr Σ. A ision prolm is lso ll lngug. Givn n instn I Σ, wht w wnt n lgorithm to o is to i whthr I L or I L. Dfinition 1.51 (Ys/no instns). Lt L Σ lngug. An instn I 1 Σ suh tht I 1 L is ll ys instn of L. An instn I 2 Σ suh tht I 2 L is ll no instn of L. In othr wors, n lgorithm tht solvs ision prolm P will for givn instn I output ys if I P n no othrwis. Dfinition 1.52 (Instn siz). Th siz of n instn I Σ is I log Σ, i.. th numr of its n to no I. Whn ling with prolms on grphs it is mor onvnint to not th siz of n instn grph G = (V, E) s V + E. For gnrl grphs this will quivlnt to Dfinition 1.52 up to smll polynomil ftor n for plnr grphs up to onstnt ftor[28]. Vry oftn w will only onsir th numr of vrtis whn tlking out th siz of grph, n for plnr grphs this will gin quivlnt up to onstnt ftor. Dision prolms r oftn rlt to wht w ll optimiztion prolms, ut s w on t n tht finition for our isussion hr, w rfr to Sipsr[27, p. 393] for mor info on this. 1.2.6 Algorithms n Runtim Dfinition 1.53 (Big-O nottion). Lt f : N N n g : N N omputl funtions. W sy tht f is O(g) if f(n) g(n), for som onstnt. This is lik sying tht f will nvr gt muh iggr thn g, n is usful for sriing runtims of lgorithms. Dfinition 1.54 (Runtim). Lt A n lgorithm tht trmins whthr givn instn I is ys or no instn to prolm P. Lt I = n, n lt th numr of stps A uss to trmin if I P oun y funtion f(n). W thn sy tht th runtim of A is O(f(n)). Algorithms hving runtim O(n ), whr is onstnt, w sy r polynomil tim lgorithms. Whn w writ poly(n) or n O(1) it is quivlnt to O(n ). Algorithms hving runtim O( n ) w ll xponntil tim lgorithms. Whn w tlk out ffiint lgorithms, w r tlking out polynomil tim lgorithms. Somtims w ll writ polynomil lgorithms n xponntil lgorithms, mning polynomil tim n xponntil tim lgorithms. 12
1.2.7 Dominting St Now lt s turn k to th Dominting St prolm w introu rlir. W strt out y fining it formlly: Dfinition 1.55 (Dominting St). A Dominting St D of grph G = (V, E) is sust of vrtis D V suh tht N[D] = V. In othr wors, vry vrtx in G must ithr in D or hv nighor in D for D to Dominting St of th grph. Th ision prolm now oms Dominting St (DS) Input: Grph G, n intgr k. Qustion: Is thr Dominting St D of G of siz t most k? In trms of th finition of ision prolms (Dfinition 1.50), this mns w hv lngug Dominting St whr vry ys instn is of th form (G, k), whr G is grph hving Dominting St of siz t most k. W will oftn ll th vrtis in givn ominting st of G for omintors. Th prolm sttmnt is prtty strightforwr, ut to fin th solution in th gnrl s is known to hr. Dominting St is on of th funmntl NP-omplt prolms. W ll hv look t wht tht mns in th nxt stion. 1.2.8 Effiint Algorithms n NP-hrnss Whn w sign n lgorithm w woul nturlly lik it to s fst, or ffiint, s possil. W on t wnt to wit forvr for th lgorithm to output n nswr to our qustion. In omplxity thory, lgorithms with polynomil runtim is usully onsir ffiint. Algorithms with n xponntil runtim r lss sirl. Th running tim of ths xponntil lgorithms inrss vry fst whn th input siz is inrs. Dision prolms for whih thr xist polynomil tim lgorithms w sy r in th omplxity lss P. All ision prolms for whih w n in polynomil tim hk if givn solution to th prolm is vli or not, w sy r in th lss NP. Not tht ll prolms in P r lso in NP, ut not nssrily th othr wy roun. An som prolms n shown to NP-hr, mning tht ll prolms in NP n ru to thm [27, p. 304]. Tht is, if you n fin n ffiint lgorithm for ths prolms, thn ll prolms in NP n solv ffiintly. If suh n NP-hr prolm itslf is in NP, it is ll NP-omplt n is onsir mong th hrst prolms in NP. It is liv, ut not provn, tht P NP, n thrfor tht ths NP-omplt prolms only hv xponntil tim lgorithms. For mor on this, s Sipsr[27]. 1.2.9 Coping with NP-hrnss Evn though w might hv givn up on fining polynomil tim lgorithms for NP-hr prolms, w still wnt to solv thm s fst s possil. Mny of th NP-hr prolms hv lot of prtil pplitions (s th fir sttions xmpl for th Dominting St prolm), n w woul lik to l to hnl instns with spil proprtis, or whn th solution siz in qustion (th prmtr k) is smll. Approximtion lgorithms tht run in polynomil tim is ommon wy of fining n nswr tht might goo nough in mny prtil ss. Nxt, w will look t on lss of prolms tht n solv quit ffiintly whn th prmtr to th prolm is smll. 13
Th prmtr is rlvnt sonry msurmnt k to th instn siz, n for ths prolms w n fin lgorithms whr th xponntil ftor no longr pns on n, ut rthr on k. 14
Chptr 2 Fix Prmtr Trtility n Krnliztion In th fil of Prmtriz Complxity w sri th running tim of n lgorithm in trms of prmtr to th prolm, in ition to th siz of prolm instn. W gt th following finitions from Cygn, Fomin, Kowlik, Lokshtnov, Mrx, Pilipzuk, Pilipzuk n Surh [8]: Dfinition 2.1 (Prmtriz prolm). [8, Df 1.1] A prmtriz prolm is lngug L Σ N, whr Σ is fix siz lpht. Givn n instn (x, k) Σ N, k is ll th prmtr of th instn. Dfinition 2.2 (Instn siz). Th siz of n instn (x, k) of prmtriz prolm is (x, k) = x + k. Dfinition 2.3 (Fix Prmtr Trtl prolm). [8, Df 1.2] Lt L prmtriz prolm, n (x, k) Σ N. L is ll Fix Prmtr Trtl (FPT) if thr xists omputl funtion f : N N, onstnt, n n lgorithm tht n trmin if (x, k) L in tim oun y f(k) (x, k). Osrv tht w giv no oun on how fst th funtion f(k) n grow, n it will in most ss n xponntil funtion. Th upsi is tht th funtion is xponntil in th prmtr k, whih vry oftn n smll, whil th totl runtim is polynomil in n. Anlogous to th lsss P n NP, th lss of prolms tht r Fix Prmtr Trtl is ll FPT. In th nxt stion w will prsnt simpl lgorithm for th prolm Vrtx Covr tht runs in FPT tim. 2.1 Fix Prmtr Trtl Algorithm for Vrtx Covr Vrtx Covr is n xmpl of prolm tht is known to FPT, n tully n solv quit ffiintly if th prmtr is smll. Dfinition 2.4 (Vrtx Covr). A Vrtx Covr S of grph G = (V, E) is sust of vrtis S V s.t. G S hs no gs. 15
Vrtx Covr (VC) Input: Grph G, n intgr k. Qustion: Is thr Vrtx Covr S of G of siz t most k? Th Vrtx Covr prolm is known to NP-hr [18], ut lukily it is lso known to Fix Prmtr Trtl. W ll now show simpl FPT lgorithm for trmining th nswr to th VC ision prolm. Th on ky osrvtion w n for th lgorithm is this: Osrvtion 2.5. For ny vrtx ovr of grph G, n for vry g of G, t lst on of its npoints must in th vrtx ovr. Figur 2.1: At lst on of th r g s npoints must in th vrtx ovr, so w try oth hois. This is th si i hin Algorithm 2.1. Algorithm 2.1 Fix Prmtr Trtl lgorithm forvrtx Covr Input: Grph G n intgr k N. Output: ys if G hs vrtx ovr of siz k, no othrwis. 1: prour solv V C (G, k) 2: if E(G) = thn 3: rturn ys 4: ls if k = 0 thn 5: rturn no 6: ls 7: Pik ny (u, v) E(G) 8: rturn solv V C (G u, k 1) solv V C (G v, k 1) 9: n if 10: n prour As w n s, th lgorithm piks ny rmining g in grph G of instn (G, k), n tris to rmov on of it s npoints, otining th grph G. Th rsulting instn (G, k 1) is smllr, so it rursivly invoks itslf on this instn. If this ll fins solution to (G, k 1), w know thr s lso solution to (G, k). If it is not, th lgorithm tris rmoving th othr npoint inst, n if it gin fils to fin solution, it n onlu thr s no solution to (G, k). W n fin th runtim of th lgorithm y osrving tht in th worst s vry invotion will rnh into two nw invotions. Sin w for vry invotion rs th prmtr y 1, th rursion tr n om t most k lvls p. Th rst of th work is ll polynomil, n th rsulting runtim is 2 k n O(1), whih is on th form f(k) poly(n) n hn is FPT. 16
2.2 Krnls Krnliztion is thniqu tht ls to FPT running tims for prmtriz prolms. Hr givn input instn is ru to n quivlnt instn hving siz oun y th givn prmtr. Th following finitions r pt from Cygn t l[8, 2.1]. Dfinition 2.6 (Equivlnt instns). Lt L prmtriz prolm. Two instns (x, k), (x, k ) Σ N r ll quivlnt if (x, k) L (x, k ) L. A prprossing lgorithm is n lgorithm tht givn n instn (I, k) to prolm moifis this instn n output n quivlnt instn (I, k ). Dfinition 2.7 (Output siz of prprossing lgorithm). Th siz of th output from prprossing lgorithm A is funtion siz A : N N : siz A (k) = sup{ I + k : (I, k ) = A(I, k), I Σ } In othr wors w look t ll possil instns with fix prmtr k to th lgorithm, n msur th output siz s th siz of th lrgst output. Th siz is onsir infinit if th siz nnot oun y funtion of k. Dfinition 2.8 (Krnliztion lgorithm, krnl). Lt L prmtriz prolm. A krnliztion lgorithm, or simply krnl, is n lgorithm A tht tks s input prolm instn (I, k) of L, n in polynomil tim rts nw quivlnt instn (I, k ) suh tht (I, k ) f(k), whr f : N N is omputl funtion not pnnt on I. siz A (k) is ll th siz of th krnl. Suh krnliztion lgorithms vry oftn pro y pplying rution ruls to th input instn. An xmpl in our fir sttion s oul town with no ros to nighoring towns. Of ours w woul hv to put fir sttion in this town, so w oul ignor tht town n rs th numr of fir sttions to put y 1, rsulting in n quivlnt, smllr instn. Anothr simpl osrvtion is shown in Figur 2.2, whr town hving t lst on nighor of gr 1, lwys is goo hoi for omintor. Thrfor it is sf to rmov ll ut on of ths gr 1 towns from th instn. Figur 2.2: Th gr 1 vrtis fors th grn vrtx to in goo ominting st. This is rgrlss of how mny suh gr 1 vrtis thr r, n w n rmov ll ut on. A prolm hving krnliztion lgorithm is quivlnt to th prolm ing fix-prmtr trtl, s th nxt two lmms will show. 17
Lmm 2.9. If prmtriz prolm P is il n mits krnl, thn it n solv in FPT tim. Proof. Assum P is il in tim g(n) y using som lgorithm A, n lt (x, k) n instn of siz n to P. Apply th polynomil tim krnliztion lgorithm tht outputs n quivlnt instn (x, k ) hving siz oun y som f(k). W n now pply A, whih in tim O(g(f(k))) = O(f (k)) outputs n nswr to th quivlnt instn, n hn to th originl instn. Th totl runtim is O(f (k) + poly(n)), whih is FPT. Using this pproh w n fin FPT lgorithms for ll prolms tht hv krnliztion lgorithm. Surprisingly, th onvrs is lso tru: Lmm 2.10. If prmtriz prolm P is solvl in FPT tim, thn it mits krnl. Th proof of Lmm 2.10 is out of sop n not nssry for this isussion, so w rfr to Cygn t l.[8]. Th two lmms togthr givs us th following usful thorm. Thorm 2.11. A prolm P is FPT if n only if it mits krnl. Th siz of th krnl is importnt whn w wnt to hiv ffiint lgorithms for solving prolm. If w n oun th siz of th krnl to som linr funtion f(k) = O(k), w sy tht th prolm mits linr krnl. Similrly, if w n oun th siz to som polynomil funtion g(k) = O(k ) for som onstnt, w sy th prolm mits polynomil krnl. Som prolms r hrr to fin smll krnls for, n w might not l to fin ny krnls ttr tht xponntil ons, h(k) = 2 O(k), or vn n xponntil towr, i(k) = 2 2 2 smll krnls s possil. O(k). Th gol for ny prolm is to fin s 2.2.1 Rution Ruls A krnliztion vry oftn invok smll, polynomil tim suroutins ll rution ruls. W will us ths svrl tims throughout th thsis. Th following finitions r from Cygn t l.[8, p.18]. Dfinition 2.12 (Rution rul). A rution rul is funtion ϕ : Σ N Σ N tht mps n instn (x, k) to n quivlnt instn (x, k ) suh tht ϕ is omputl in tim polynomil in x n k. Dfinition 2.13 (Sounnss of rul). Th proprty of rution rul tht it trnslts n instn to n quivlnt on, is ll th sounnss of th rul. Whn w rt rution ruls, w will lso prov thir sounnss, tht is to show tht th ruls will prou quivlnt instns. 18
Prt II A Linr Krnl for Plnr Dominting St 19
Chptr 3 Plnr Dominting St W know Dominting St is n NP-omplt prolm, so th nxt nturl qustion to sk might if it is FPT. In th gnrl s it s unlikly to, s tht woul imply omplxity thorti rsult onsir lmost s unlikly s P = NP. W rfr to Cygn t l.[8, h.13] for mor on this. But on plnr grphs it is known to FPT. It vn mits linr krnl. In [2] Alr, Fllows & Nirmir introu th novl thniqu of Rgion Domposition for giving krnls for prolms on plnr grphs. Using st of rution ruls n this thniqu thy hiv krnl for Plnr Dominting St with 335k vrtis. Ltr Chn, Frnu, Knj, n Xi[6] improv on th rution ruls y Alr t l., n wr l to show krnl uppr oun y 67k, whih is th urrnt smllst krnl known for this prolm. In th following stion w giv finitions tht lssify th vrtis insi th nighorhoo of vrtx, s this is somthing w will mk us of frquntly. 3.1 Nighorhoo of Vrtx Th following finitions r opt from Alr t l.[2], n w will us thm throughout th ppr. Lt G = (V, E) grph. Dfinition 3.1 (Nighorhoo of vrtx). Lt v V. W ivi th nighorhoo of v into 3 isjoint sts: N 1 (v) = {u N(v) N(u) \ N[v] } N 2 (v) = {u N(v) \ N 1 (v) N(u) N 1 (v) } N 3 (v) = N(v) \ (N 1 (v) N 2 (v))) In othr wors, th vrtis N 1 (v) N(v) r thos vrtis hving t lst on nighor outsi th nighorhoo of v, N 2 (v) N(v) r thos hving t lst on nighor in N 1 (v), n N 3 (v) N(v) r th rst of v s nighors. 21
v Figur 3.1: Th nighorhoo of vrtx v. Vrtis from N 1 (v) r olor grn, N 2 (v) r olor r, n N 3 (v) r olor lu. 3.2 Rution Ruls Alr t l introu two rution ruls in thir ppr. W won t n thm in our isussion, so w rfr to thir ppr for sription n proof of sounnss [2, Rul 1, Rul 2]. Howvr, w prsnt two simpl rution ruls hr, inspir y thir first rul. Lt G = (V, E): Rution Rul 3.1. Lt x, y N 2 (v) N 3 (v) for som vrtx v V n (x, y) E, thn rmov th g (x, y). y y x x v v Figur 3.2: W n mk N 2 (v) N 3 (v) vrtis inpnnt. In this sitution v is oviously ttr hoi for ominting th vrtis in its nighorhoo thn x n y. This is why w n rmov th g twn x n y, sin non of thm will hosn s omintors. W now prov th sounnss of this rul. Lmm 3.2. Rution Rul 3.1 is soun. Proof. Lt D ominting st of siz k in G, n lt G th grph otin ftr prforming th rution rul. By th finition of N 2 (v) n N 3 (v), w know tht N[x] N[v] n N[y] N[v], n hn v woul lwys omintor t lst s goo s ithr of x, y. Thrfor w n sfly ssum tht nithr x nor y is in D. Now D is lso ominting st of siz k in G. For th othr irtion, lt D ominting st of siz k in G. D will lso ominting st of siz k in G, sin ing n g to th grph nnot mk D non-ominting. 22
Rution Rul 3.1 mks N 2 (v) N 3 (v) n inpnnt st, whih us usful for ltr nlysis. Also not tht th rul n rri out in polynomil tim. Rution Rul 3.2. Lt x, y N 2 (v) N 3 (v) for som vrtx v V n N(x) N(y), thn rmov y from th grph. y x x v v Figur 3.3: W n mk rmov vrtis from N 2 (v) N 3 (v) tht only srv th funtion of ing omint. Th intuition hin Rul 3.2 is tht non of x, y will usful s omintors, n thrfor tht th only wy thy fft th siz of th ominting st is tht thy n to omint. But sin y is omint th momnt x is, w n rmov y. Lmm 3.3. Rution Rul 3.2 is soun. Proof. Lt D ominting st of siz k in G, n lt G th grph otin ftr prforming th rution rul. W n ssum (s th proof of Lmm 3.3) tht nithr x nor y is in D. Now D is lso ominting st of siz k in G. For th othr irtion, lt D ominting st of siz k in G. D will omint x, n sin N[x] N[v], w n ssum x D (if it ws, putting v in inst woul t lst s goo). In G w know tht N(x) N(y), n D will lso ominting st of siz k in G. Th fft of th two rution ruls is tht N 2 (v) N 3 (v) will inpnnt, n tht no vrtx from this st will hv nighorhoo tht is suprst of nothr vrtx from th st. Not tht in th s w hv t lst on N 3 (v) vrtx, Rul 3.1 will ru this to gr 1 vrtx, n ny othr vrtx from th N 2 (v) N 3 (v) will rmov y Rul 3.2. 3.3 Rgions n Rgion Domposition s Trt y Alr t l. Aftr prforming th rution ruls on th input instn, w wnt to uppr oun th siz of th rsulting, quivlnt, instn. This is whr Rgion Domposition oms in, y iviing th grph into svrl rgions n thn uppr ouning th numr of vrtis outsi th rgions, uppr ouning th numr of suh rgions, n t lst uppr ouning th numr of vrtis insi suh rgion. In this stion w will sri rgions n rgion omposition s fin y Alr t l. [2], n point out th prolm with ths finitions. This shoul giv n insight into why w wnt to rfin ths onpts, n whih spil ss w must mk sur to hnl in our nw finitions. 23
Givn ru grph G, lt k = D th siz of Dominting St D in this grph. Alr t l. fin Rgions n Rgion Domposition s follows: Dfinition 3.4 ([2, Dfinition 2]). Lt G = (V, E) pln grph. A rgion R(v, w) twn two vrtis v, w is los sust of th pln with th following proprtis: 1. th ounry of R(v, w) is form y two simpl pths P 1 n P 2 in V whih onnt v n w, n th lngth of h pth is t most thr (gs), n 2. ll vrtis whih r stritly insi (i.. lying in th rgion, ut not sitting on th ounry) th rgion R(v, w) r from N(v, w). For rgion R = R(v, w), lt V (R) not th vrtis longing to R, i.., V (R) := {u V u sits insi or on th ounry of R} Dfinition 3.5 ([2, Dfinition 3]). Lt G = (V, E) pln grph n D V. omposition of G is st R of rgions twn pirs of vrtis in D suh tht 1. for R(v, w) R no vrtx from D (xpt for v, w) lis in V (R(v, w)) n A D-rgion 2. no two rgions R 1, R 2 R o intrst (howvr, thy my touh h othr y hving ommon ounris). For D-rgion omposition R w fin V (R) := R R V (R). A D-rgion omposition is ll mximl if thr is no rgion R R suh tht R := R {R} is D-rgion omposition with V (R) V (R ). Dfinition 3.6 ([2, Dfinition 4]). Th inu grph G R = (V R, E R ) of D-rgion omposition R of G is th grph with possil multipl gs whih is fin y V R := D n E R := {{v, w} thr is rgion R(v, w) R twn v, w D}. Proposition 3.7 ([2, Proposition 1]). For ru pln grph G with ominting st D, thr xists mximl D-rgion omposition R suh tht G R is thin. W on t stt thir finition of thin hr, ut for grph to thin, it hs to plnr, mong othr things. Alr t l. lso show how to fin suh mximl D-rgion omposition of pln grph, n thy us this togthr with Dfinition 3.6 n Proposition 3.7 to prov tht suh omposition will hv t most 3k rgions. This rquir th omposition grph G R to plnr, if not Proposition 3.7 woul not hol. For G R to plnr, two rgions n not llow to ross in th rgion omposition, tht is thir rwing in th pln n t ross. This must lso hol vn whn oth r gnrt. Hr gnrt rgion R(v, w) mns rgion whr th pths P 1 n P 2 shr t lst on ommon vrtx in ition to v, w. Togthr with proofs ouning th numr of vrtis insi n outsi of rgions in mximl D-rgion omposition, this givs ris to krnl of linr siz. Howvr, thr is prolm with th proof ouning th numr of vrtis outsi rgions. In th proof of [2, Lmm 6], th uthors us tht thr r no rossing rgions in rgion omposition to prov tht ll vrtis from N 1 (v) r insi rgions. But to omplish this thy sy tht for th gnrt rgion R = {v, u, u, w} to ross in R without u lry ing in rgion, th g {u, w} must ross rgion R(x, y) R, whih implis tht w is on th ounry or insi th rossing rgion. But onsir th s whr R(x, y) is th gnrt rgion {x, u, y}, s shown in Figur 3.4. From Dfinition 3.5 it is hr to xtrt whthr two suh gnrt rgions r onsir rossing in this s. If w onsir thm rossing thn R(x, y) n ross R without w ing insi or on th ounry of R(x, y), mking th proof of [2, Lmm 6] invli. If th rgions r not onsir rossing thn th omposition grph from Dfinition 3.6 is no longr plnr, whih is ruil for Proposition 3.7. 24
x w u y u v Figur 3.4: Th gnrt rgion R(x, y) n ross th gnrt rgion R(v, w) without w ing on th ounry of R(x, y). W will show tht thir omposition of th grph into rgions is orrt up to minor moifitions. Ths moifitions r inonsquntil for th us of th omposition for ltr proofs. This mns tht th krnl siz vntully otin y Alr t l. is orrt, whih w will onlu in th nxt hptrs. 25
26
Chptr 4 Domposing th Grph W will now mk nw finitions nssry for th isussion to follow. In Chptr 5 w will show tht minor moifitions to th thorms stt y Alr t l. r suffiint to rsolv th rrors in th proofs s isuss in Chptr 3. In Chptr 6 w will furthr lowr th krnl siz uppr oun of Chn t l.[6] y n improv nlysis. 4.1 Egs in Pln Emings Dfinition 4.1 (Eg istn). Lt (G, E) pln grph, lt v V (G), n 1, 2 E(G) two gs oth inint to v. Strting from th point whr th polygon lin of 1 intrsts th unit isk of v in th ming, mov lokwis long th isk. Lt k th numr of polygon lins nountr for nountring th polygon lin of 2. Do th sm, now strting from 2 inst, lt l th numr of polygon lins nountr for nountring 1. Th g istn with rspt to v of th two gs 1, 2, is not ist v ( 1, 2 ) = min(k, l). u Figur 4.1: Th g istn of gs n with rspt to u is 1. Dfinition 4.2 (Consutiv gs). Egs hving n g istn of 0 with rspt to v V, r si to onsutiv t v. u Figur 4.2: Consutiv pirs of gs t u r (, ), (, ), (, ), (, ) n (, ). 27
Dfinition 4.3 (Prlll gs). W sy tht two gs 1, n E twn vrtis v, w V (G) in multigrph G r prlll if thr xists squn of gs 1,..., n suh tht i n i+1 r onsutiv t oth u n v for ll i {1,..., n 1}. v 1 v 2 Figur 4.3: Th r gs r prlll gs. So r th lu gs. Osrv tht th grn g is not prlll to th r gs sin thy r not onsutiv t v 2. 4.2 Wlks in Pln Emings In th following finitions, lt (G = (V, E), E = (U, P, f, g)) pln grph. Dfinition 4.4 (Join of gs). Lt 1, 2 E two gs oth inint to th vrtx v V, n lt = f(v) g( 1 ), = f(v) g( 2 ) th two points whr th gs intrst with th vrtx in th ming. Th join of 1 n 2 t v, is th linsgmnt L(, ). W writ join( 1, v, 2 ) 1 v 2 Figur 4.4: Egs 1 n 2 r oth inint to som vrtx v, n intrst with th unit isk of v in points n, mrk with r on th lft figur. join( 1, v, 2 ) is th linsgmnt twn n, olor in r on th right figur. Dfinition 4.5 (Drwing of wlk). Lt W = (v 1, 1, v 2, 2,..., v k, k, v k+1 ) wlk in G. W otin th rwing of W, not rw(w ), y tking th union of th points in g() for vry { 1,..., k }, togthr with th join of vry i, i+1 t v i+1, i.. ( ) rw(w ) = g( 1 ) g( i ) join( i 1, v i, i ) i {2,...,k} 28
f g f g h h Figur 4.5: A grph (lft), n wlk in this grph shown in r (right). Dfinition 4.6 (Crossing wlks). W sy tht two wlks W 1, W 2 in G ross if rw(w 1 ) rw(w 2 ). Dfinition 4.7 (Simpl yli wlk). A simpl yli wlk C is yli wlk W = (v 1, 1,..., v k, k, v 1 ) in pln grph suh rw(c) = rw(w ) join( 1, v 1, k ) is simpl polygon. W rfr to rw(c) s th rwing of C. Sin w will only r out simpl yli wlks, w will oftn just rfr to thm s yli wlks. Dfinition 4.8 (Intrior of yli wlk). Lt C yli wlk in pln grph. Th intrior f of rw(c) is ll th intrior of th yli wlk, whil th xtrior f is ll th xtrior of th yli wlk. Dfinition 4.9 (Vrtis in yli wlk). Th vrtis rprsnt y unit isks ing stritly in th intrior of yli wlk, r ll intrnl vrtis of th wlk. Th vrtis rprsnt y unit isks ing stritly in th xtrior of yli wlk, r ll xtrnl vrtis. Th rst, hving thir unit isks intrst with th rwing of th wlk, r ll vrtis on th wlk. Osrv tht y Dfinition 4.9, th vrtis on th wlk C = (v 1, 1,..., v k, k, v 1 ) r xtly th vrtis {v 1,..., v k }. 29
f g h Figur 4.6: Th rwing of yli wlk shown in r. h is th only intrnl vrtx of th wlk, whil f is th only xtrnl vrtx.,,,,, g r vrtis on th wlk. Dfinition 4.10 (Ar of yli wlk). Th r of yli wlk C, not r(c) is th union of th points on th wlk n ll points in th intrior. Dfinition 4.11 (Crossing yli wlks). W sy tht two yli wlks C 1, C 2 ross if thir rs intrst, i.. r(c 1 ) r(c 2 ). f g h Figur 4.7: Th two yli wlks shown in r n yllow ross sin thir rs intrst. Dfinition 4.12 (Cross t vrtx). W sy tht two yli wlks C 1, C 2 ross t vrtx v if rw(c 1 ) rw(c 2 ) f(v). Dfinition 4.13 (Cross t g). W sy tht two yli wlks C 1, C 2 ross t n g if rw(c 1 ) rw(c 2 ) g(). W will us th nxt wll-known ft without proof: Ft 4.14. Lt P 1 n P 2 two simpl polygons, n lt A 1 n A 2 thir intrior fs, rsptivly. If A 1 A 2, thn P 1 P 2. Lmm 4.15. Lt C 1, C 2 two yli wlks tht ross. Thn, ithr 30
r(c 1 ) r(c 2 ) r(c 2 ) r(c 1 ) C 1 n C 2 ross t som vrtx v Proof. Sin C 1 n C 2 ross, thn y finition r(c 1 ) r(c 2 ). This n hppn if ithr yl s r is th sust of th othr, s in th first two ss. For th thir s, ssum tht nithr of th first two ss pply. In tht s w must hv rw(c 1 ) rw(c 2 ) y Ft 4.14, whih mns thy ithr ross t n g, or ross t som vrtx. Assum tht thy ross t n g = (v 1, v 2 ). Thn y finition of yli wlk, v 1 n v 2 will vrtis on oth wlks, n th two wlks will ross t point in oth of ths vrtis, nmly g() f(v 1 ) n g() f(v 2 ). Osrvtion 4.16. If r(c 1 ) r(c 2 ) g(), whr C 1 n C 2 r yli wlks n = (v 1, v 2 ), thn r(c 1 ) r(c 2 ) f(v 1 ) n r(c 1 ) r(c 2 ) f(v 2 ) s wll. Lmm 4.17. Lt C 1 = (v 1, 1,..., v i, i, v i+1, i+1, v i..., v k, k, v 1 ) n C 2 yli wlks suh tht i n i+1 r prlll gs going twn v i n v i+1, n C 1 n C 2 on t ross t v i. Thn C 1 n C 2 nnot ross t v i+1. Proof. Sin C 1 n C 2 on t ross t v i, r(c 1 ) r(c 2 ) g( i ) = n r(c 1 ) r(c 2 ) g( i+1 ) =. By Osrvtion 4.16 this mns tht for C 1 n C 2 to ross t v i+1, J = join( i, v i+1, i+1 ) woul n to ross C 2 t v i+1. For this to hppn thr woul n to xist join J = join( 1, v i+1, 2) s.t. J J, whr 1 i, 2 i, 1 i+1, 2 i+1. Lt th linsgmnts rprsnting th joins L(, ) = J n L(, ) = J. For J to intrst J, n woul n to sprt n on th orr of f(v i+1 ), whih ontrits tht i n i+1 r onsutiv, s Figur 4.8. i i+1 vi v i+1 Figur 4.8: Two joins rossing ontrits tht i n i+1 is onsutiv t v i+1. Dfinition 4.18 (Contntion of simpl wlks). Lt W 1 = (v1, 1 1 1,..., 1 k 1, v1 k ) n W 2 = (v1, 2 2 1,..., 2 l 1, v2 l ) two simpl wlks in G suh tht rw(w 1) rw(w 2 ) =, n v1 1 = v1 2 n vk 1 = v2 l. W sy th ontntion of W 1 n W 2 is th yli wlk hving th rwing rw(w 1 ) rw(w 2 ) join( 1 1, v1, 1 2 1) join( 1 k 1, v1 k, 2 l 1 ). 31
f g f g h h Figur 4.9: Contntion of two simpl wlks tht rsults in yli wlk. 4.3 Rgions Dfinition 4.19 (Rgion). Lt (G, E) pln grph, n lt v, w V (G) two vrtis in G. A rgion R = R(v, w) twn v n w is fin y two simpl, non-rossing wlks W 1, W 2 of lngth t most 3 twn v n w, n st V (R) V (G) suh tht Th ontntion C R of W 1 n W 2 is simpl yli wlk Th vrtis on C R r not δ(r) Th intrnl vrtis of C R r not I(R) V (R) = I(R) δ(r) \ {v, w} V (R) N(v, w) C R is ll th ounry of R. δ(r) r th ounry vrtis of R. I(R) r ll th intrnl vrtis of R. V (R) r ll th vrtis longing to R. v, w r ll th npoints of th rgion. Noti tht th two wlks W 1, W 2 uniquly fin whih vrtis r in δ(r) n I(R), n hn V (R). v w Figur 4.10: Th rgion R = R(v, w) hs th npoints v n w. Th vrtis δ(r) \ {v, w} r olor lu, n th intrnl vrtis I(R) yllow. Ths fiv vrtis fin th st V (R). Dfinition 4.20 (Ar of rgion). Th r of rgion R, r(r), is th r of its ounry, r(c R ). Dfinition 4.21 (Crossing rgions). W sy tht two rgions R 1, R 2 ross if thir rs intrst, tht is r(r 1 ) r(r 2 ). 32
Sin th r of rgion is fin y its yli wlk, w will us Dfinition 4.12, Dfinition 4.13, Lmm 4.15 n Lmm 4.17 on rgions ltr, rfrring to thir ounry. By finition, rgions s thos sn Figur 4.11 will onsir rossing. Tht is lso th s for Figur 4.11, s th two rgions shr n g. Howvr, in th lttr s w woul lik rgions to l to touh h othr in this wy without rossing, without llowing th rgions in ) to o tht. This is th motivtion for introuing grph nrihmnts, whih w will look t in th following stion. Figur 4.11: Crossing rgions ) n ). 4.4 Grph Enrihmnt Dfinition 4.22 (Copying n g). Hving pln multigrph G = (G = (V, E), E), w fin opying n g E s th oprtion of ing nw g to th grph, suh tht n r prlll. Whn opying th g E to, w st ϵ = ϵ/2, n G = (V, E { }) n hv th sm ming s G, only ing th polygon lin of in istn ϵ from th polygon lin of. Th rsult is nw pln grph G = (G, E ) ing struturlly qul to G, ut hving on xtr g. Dfinition 4.23 (Grph nrihmnt). Lt G pln grph. W ll th grph G n nrihmnt of G if it n otin from G only using th oprtion of opying gs. G G Figur 4.12: G is n nrihmnt of G. 4.5 Rgion Domposition With th onpt of grph nrihmnts in pl, w n fin rgion omposition of pln grph. 33