A 43k Kernel for Planar Dominating Set using Computer-Aided Reduction Rule Discovery

Transcription

1 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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3 I think vryon shoul lrn how to progrm omputr us it ths you how to think - Stv Jos

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5 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.

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7 Contnts I Introution n prliminris 1 1 Introution Bkgroun n Thsis Ovrviw Trminology n Prliminris Mthmtil Nottion Points in th Pln Grphs Pln Grphs Dision Prolms Algorithms n Runtim Dominting St Effiint Algorithms n NP-hrnss Coping with NP-hrnss Fix Prmtr Trtility n Krnliztion Fix Prmtr Trtl Algorithm for Vrtx Covr Krnls Rution Ruls II A Linr Krnl for Plnr Dominting St 19 3 Plnr Dominting St Nighorhoo of Vrtx Rution Ruls Rgions n Rgion Domposition s Trt y Alr t l Domposing th Grph Egs in Pln Emings Wlks in Pln Emings Rgions Grph Enrihmnt Rgion Domposition Mximl Rgion Domposition A Linr Krnl Th Krnl of Alr t l A Smllr Krnl 43 i

8 6.1 Vrtx Flipping Uppr Boun Outsi Rgions Bouning Rgion Siz An Improv Krnl Uppr Boun III Computr-Ai Rution of Rgions 49 7 A 43k krnl for Plnr Dominting St Dfinitions A 43k Krnl Computr-Ai Rution of Dth Rgions Mor Dfinitions Rution Ruls Insi Rgion Boun Insi Innr n Singl Rgions Innr rgions Bouning th Siz of n Innr Qusi-Rgion Singl Qusi-Rgions Outr Rgions Without Possil Domintors Bounry Sizs n Vrtis Splitting Rgions Into Smllr Prts Fully Enumrt Rprsnttiv Sts Orr of Enumrtion Enumrtion of Fully Enumrt Rprsnttiv Sts Innr rgions Enumrting Singl Rgions (2, 1) s -rgions (2, 2) s -rgions (2, 2) s (1, )(,2)-rgions (3, 2) s -rgions (3, 2) s (1,2,3)(1, )-rgions (3, 3) s (1,2,3)(1,2,3)-rgions (2, 1) n -rgions (2, 2) s,n -rgions (2, 1)-rgions (2, 2)-rgions (3, 1)-rgions (3, 2)-rgions (3, 3)-rgions Implmnttion n Rsults Conlusions Opn Prolms Biliogrphy 92 ii

9 Prt I Introution n prliminris 1

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11 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

12 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

13 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

14 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 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

15 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

16 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 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

17 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

18 ɛ Figur 1.7: A plnr grph n lwys rwn with gs sprt y istn ϵ > 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

19 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

20 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 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 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 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

21 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 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] 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

22 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

23 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

24 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

25 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

26 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 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 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) = smll krnls s possil. O(k). Th gol for ny prolm is to fin s 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

27 Prt II A Linr Krnl for Plnr Dominting St 19

28

29 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

30 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

31 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 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

32 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

33 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

34 26

35 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

36 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 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

37 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

38 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 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 Lt C 1, C 2 two yli wlks tht ross. Thn, ithr 30

39 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 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 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

40 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

41 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