Stright-lin Gri Drwings o 3-Connt 1-Plnr Grhs M. Jwhrul Alm 1, Frnz J. Brnnurg 2, n Sthn G. Koourov 1 1 Drtmnt o Comutr Sin, Univrsity o Arizon, USA {mjlm, koourov}@s.rizon.u 2 Univrsity o Pssu, 94030 Pssu, Grmny rnn@inormtik.uni-ssu. Astrt. A grh is 1-lnr i it n rwn in th ln suh tht h g is ross t most on. In gnrl, 1-lnr grhs o not mit strightlin rwings. W show tht vry 3-onnt 1-lnr grh hs stright-lin rwing on n intgr gri o urti siz, with th xtion o singl g on th outr tht hs on n. Th rwing n omut in linr tim rom ny givn 1-lnr ming o th grh. 1 Introution Sin Eulr s Königsrg rig rolm ting k to 1736, lnr grhs hv rovi intrsting rolms in thory n in rti. Using th lort thnius o nonil orring n Shnyr rlizrs, vry lnr grh n rwn on gri o urti siz, n suh rwings n omut in linr tim [15, 21]. Th r oun is symtotilly otiml, sin th nst tringl grhs r lnr grhs n ruir Ω(n 2 ) r [10]. Th rwing lgorithms wr rin to imrov th r ruirmnt or to mit onvx rrsnttions, i.., whr h innr is onvx [5, 8] or stritly onvx [1]. Howvr, most grhs r nonlnr n rntly, thr hv n mny ttmts to stuy lrgr lsss o grhs. O rtiulr intrst r 1-lnr grhs, whih in sns r on st yon lnr grhs. Thy wr introu y Ringl [20] in n ttmt to olor lnr grh n its ul. Although it is known tht 3-onnt lnr grh n its ul hv stright-lin 1-lnr rwing [24] n vn on urti gri [13], littl is known out gnrl 1-lnr grhs. It is NP-hr to rogniz 1- lnr grhs [16, 18] in gnrl, lthough thr is linr-tim tsting lgorithm [11] or mximl 1-lnr grhs (i.., whr no itionl g n without violting 1-lnrity) givn th th irulr orring o inint gs roun h vrtx. A 1-lnr grh with n vrtis hs t most 4n 8 gs [4, 14, 19] n this ur oun is tight. On th othr hn stright-lin rwings o 1-lnr grhs my hv t most 4n 9 gs n this oun is tight [9]. Hn not ll 1-lnr grhs mit stright-lin rwings. Unlik lnr grhs, mximl 1-lnr grhs n muh srsr with only 2.64n gs [6]. 1 Rsrh suort in rt y NSF grnts CCF-1115971 n DEB 1053573. 2 Rsrh suort y th Dutsh Forshungsgminsht, DFG, grnt Br 835/18-1.
i 11 10 j 9 g 8 7 6 h h 5 4 3 j g g 2 1 i h 0 1 0 1 2 3 4 5 6 7 8 9 10 () () () 8 7 6 5 4 3 2 1 0 h g 0 1 2 3 4 5 6 7 8 () Fig. 1. () () A 3-onnt 1-lnr grh n its stright-lin gri rwing (with on n in on g), () () nothr 3-onnt 1-lnr grh n its stright-lin gri rwing. Thomssn [23] rrs to 1-lnr grhs s grhs with ross inx 1 n rov tht n m 1-lnr grh n turn into stright-lin rwing i n only i it xlus B- n W -onigurtions; s Fig. 2. Ths orin onigurtions wr irst isovr y Egglton [12] n us y Hong t l. [17], who show tht th onigurtions n tt in linr tim i th ming is givn. Thy lso rov tht thr is linr tim lgorithm to onvrt 1-lnr ming without B- n W - onigurtions into stright-lin rwing, ut without ouns or th rwing r. In this r w sttl th stright-lin gri rwing rolm or 3-onnt 1- lnr grhs. First w omut norml orm or n m 1-lnr grh with no B-onigurtion n t most on W -onigurtion on th outr. Thn, tr ugmnting th grh with s mny lnr gs s ossil n thn lting th rossing gs, w in 3-onnt lnr grh, whih is rwn with stritly onvx s using n xtnsion o th lgorithm o Chrok n Knt [8]. Finlly th irs o rossing gs r rinsrt into th onvx s. This givs stright-lin rwing on gri o urti siz with th xtion o singl g on th outr, whih my n on n (n this xtion is unvoil); s Fig. 1. In ition, th rwing is otin in linr tim rom givn 1-lnr ming. 2 Prliminris A rwing o grh G is ming o G into th ln suh tht th vrtis r m to istint oints n h g is Jorn r twn its noints. A rwing is lnr i th Jorn rs o th gs o not ross n it is 1-lnr i h g is ross t most on. Not tht rossings twn gs inint to th sm vrtx r not llow. For xml, K 5 n K 6 r 1-lnr grhs. An ming o grh is lnr (rs. 1-lnr) i it mits lnr (rs. 1-lnr) rwing. An ming siis th s, whih r toologilly onnt rgions. Th unoun is th outr. A in lnr grh is sii y yli sun o gs on its ounry (or uivlntly y th yli sun o th noints o th gs). Aoringly, 1-lnr ming E(G) siis th s in 1-lnr rwing o G inluing th outr. A 1-lnr ming is witnss or 1-lnrity. In rtiulr, E(G) sris th irs o rossing gs n th s whr th gs
() () () Fig. 2. () An ugmnt X-onigurtion, () n ugmnt B-onigurtion, () n ugmnt W -onigurtion. Th grhs inu y th soli gs r ll n X-onigurtion (), B-onigurtion (), n W -onigurtion (). ross n hs linr siz. Eh ir o rossing gs (, ) n (, ) inus rossing oint. Cll th sgmnt o n g twn th vrtx n th rossing oint hlg. Eh hl-g is imrml, nlogous to th gs in lnr rwings, in th sns tht no g n ross suh hl-g without violting th 1-lnrity o th ming. Th non-ross gs r ll lnr. A lnriztion G is otin rom E(G) y using th rossing oints s rgulr vrtis n rling h rossing g y its two hl-gs. A 1-lnr ming E(G) n its lnriztion shr uivlnt mings, n h is givn y list o gs n hl-gs ining it, or uivlntly, y list o vrtis n rossing oints o th gs n hl gs. Egglton [12] ris th rolm o rognizing 1-lnr grhs with rtilinr rwings. H solv this rolm or outr-1-lnr grhs (1-lnr grhs with ll vrtis on th outr-yl) n roos thr orin onigurtions. Thomssn [23] solv Egglton s rolm n hrtriz th rtilinr 1-lnr mings y th xlusion o B- n W-onigurtions; s Fig. 2. Hong t l. [17], otin similr hrtriztion whr th B- n W -onigurtions r ll th Bulgri n Gui grhs. Thy lso show tht ll ourrns o ths onigurtions n omut in linr tim rom givn 1-lnr ming. Dinition 1. Consir 1-lnr ming E(G): A B-onigurtion onsists o n g (, ) n two gs (, ) n (, ) whih ross in som oint suh tht n li in th intrior o th tringl (,, ). Hr (, ) is ll th s o th onigurtion. An X-onigurtion onsists o ir (, ) n (, ) o rossing gs whih os not orm B-onigurtion. A W-onigurtion onsists o two irs o gs (, ), (, ) n (, ), (, ) whih ross in oints n, suh tht,,, li in th intrior o th urngl,,,. Hr gin th g (, ), i rsnt is th s. Osrv tht or ll ths onigurtions th s gs my ross y nothr g, whrs th rossing gs r imrml; s Fig 2. Thomssn [23] n Hong t l. [17] rov tht or 1-lnr ming to mit stright-lin rwing, B- n W -onigurtions must xlu:
Proosition 1. A 1-lnr ming E(G) mits stright-lin rwing with toologilly uivlnt ming i n only i it os not ontin B- or W -onigurtion. Augmnt givn 1-lnr ming E(G) y ing s mny gs to E(G) s ossil so tht G rmins siml grh n th nwly gs r lnr in E(G). W ll suh n ming lnr-mximl ming o G n th ortion lnr-mximl ugmnttion. (Not tht Hong t l. [17] olor th lnr gs o 1-lnr ming r n ll lnr-mximl ugmnttion r ugmnttion.) Th lnr sklton P(E(G)) onsists o th lnr gs o lnr-mximl ugmnttion. It is lnr m grh, sin ll irs o rossing gs r omitt. Not tht th lnr ugmnttion n th lnr sklton r in or n ming, not or grh. A grh my hv irnt mings whih giv ris to irnt onigurtions n ugmnttions. Th notion o lnr-mximl ming is irnt rom th notions o mximl 1-lnr mings n mximl 1-lnr grhs, whih r suh tht th ition o ny g violts 1-lnrity (or simliity) [6]. Th ollowing lim, rovn in mny rlir rs [6, 14, 17, 22, 23], shows tht rossing ir o gs inus K 4 in lnr-mximl ming, sin missing gs o K 4 n without inuing nw rossings. Lmm 1. Lt E(G) lnr-mximl 1-lnr ming o grh G n lt (, ) n (, ) two rossing gs. Thn th our vrtis {,,, } inu K 4. By Lmm 1, or lnr-mximl ming h X-, B-, n W -onigurtion is ugmnt y itionl gs. Hr w in ths ugmnt onigurtions. Dinition 2. Lt E(G) lnr-mximl 1-lnr ming o grh G. An ugmnt X-onigurtion onsists o K 4 with vrtis (,,, ) suh tht th gs (, ) n (, ) ross insi th urngl. An ugmnt B-onigurtion onsists o K 4 with vrtis (,,, ) suh tht th gs (, ) n (, ) ross yon th ounry o th urngl. An ugmnt W-onigurtion onsists o two K 4 s (,,, ) n (,,, ) on o whih is in n ugmnt X-onigurtion n th othr in n ugmnt B-onigurtion. For n ugmnt X- or ugmnt B-onigurtion, th gs not inuing rossing with othr gs in th onigurtion in yl, w ll it th sklton. In h onigurtion, th gs on th outr-ounry o th m onigurtion n not inuing rossing with othr gs in th onigurtion r th s gs. Using th rsults o Thomssn [23] n Hong t l. [17], w n now hrtriz whn lnr-mximl 1-lnr ming o grh mits stright-lin rwing: Lmm 2. Lt E(G) lnr-mximl 1-lnr ming o grh G. Thn thr is stright-lin 1-lnr rwing o G with toologilly uivlnt ming s E(G) i n only i E(G) os not ontin n ugmnt B-onigurtion. Proo. Assum E(G) ontins n ugmnt B-onigurtion. Thn it ontins B- onigurtion n hs no stright-lin 1-lnr rwing y Proosition 1. Convrsly, i E(G) hs no stright-lin 1-lnr rwing thn y Proosition 1 it ontins t lst on B- or W -onigurtion. Sin Γ is lnr-mximl ming, y Lmm 1 h rossing g ir in E(G) inus K 4. Thus th ott gs in Fig. 2() () must rsnt in ny B- or W - onigurtion, inuing n ugmnt B-onigurtion.
Th norml orm or n m 1-lnr grh E(G) is otin y irst ing th our lnr gs to orm K 4 or h ir o rossing gs whil routing thm losly to th rossing gs n thn rmoving ol ulit gs i nssry. Suh n ming o 1-lnr grh is norml ming o it. A norml lnrmximl ugmnttion or n m 1-lnr grh is otin y irst ining norml orm o th ming n thn y lnr-mximl ugmnttion. Lmm 3. Givn 1-lnr ming E(G), th norml lnr-mximl ugmnttion o E(G) n omut in linr tim. Proo. First ugmnt h rossing o two gs (, ) n (, ) to K 4, suh tht th gs (, ), (, ), (, ), (, ) r n in s o ulit th ormr g is rmov. Thn ll ugmnt X-onigurtions r mty n ontin no vrtis insi thir skltons. Nxt tringult ll s whih o not ontin hl-g, rossing g, or rossing oint. Eh st n on in linr tim. 3 Chrtriztion o 3-Connt 1-Plnr Grhs Hr w hrtriz 3-onnt 1-lnr grhs y norml ming, whr th rossings r ugmnt to K 4 s suh tht th rsulting ugmnt X-onigurtions hv vrtx-mty skltons n thr is no ugmnt B-onigurtion xt or t most on ugmnt W-onigurtion with ir o rossing gs in th outr. Lt E(G) 1-lnr ming o grh G. Eh ir o rossing gs inus rossing oint n th rossing gs n thir hl-gs r imrml s thy nnot ross y othr gs without violting 1-lnrity. An imrml th in E(G) is n intrnlly-isjoint sun P = v 1, 1, v 2, 2,..., v n, n, v n+1, whr v 1, v 2,..., v n+1 r (rgulr) vrtis o G, 1, 2,..., n r rossing oints in E(G) n (v i, i ), ( i, v i+1 ) or h i {1, 2,..., n} r hl gs. I v n+1 = v 0, thn P is n imrml yl. An imrml yl is srting whn it hs vrtis oth insi n outsi o it, sin lting its vrtis isonnts G. Lmm 4. Lt G = (V, E) 3-onnt 1-lnr grh with lnr-mximl 1-lnr ming E(G). Thn th ollowing onitions hol. A. (i) Two ugmnt B-onigurtions or two ugmnt X-onigurtions nnot on th sm si o ommon s g. (ii) Suos n ugmnt B-onigurtion B n n ugmnt X-onigurtion X r on th sm si o ommon s g (, ). Lt n th rossing oints or X n B, rstivly n lt R(X) n R(B) th rgions insi th skltons o X n B. Thn ll vrtis o V \ {, } r insi th imrml yl i R(X) R(B); othrwis ll vrtis o V \ {, } r outsi th imrml yl. B. (i) I two ugmnt B-onigurtions r on oosit sis o ommon s g (, ), with rossing oints n, rstivly, thn ll th vrtis o V \ {, } r insi th imrml yl.
() () () () () () (g) Fig. 3. Illustrtion or th roo o Lmm 4. (ii) I two ugmnt X-onigurtions r on oosit sis o ommon s g (, ), with rossing oints n, rstivly, thn ll th vrtis o V \ {, } r outsi th imrml yl. (iii) An ugmnt B-onigurtion n n ugmnt X-onigurtion nnot shr ommon s g rom oosit sis. Proo. Conition A.(i) n B.(iii) hol us h o ths onigurtions inus srting imrml yl in E(G) with only two (rgulr) vrtis rom G, ontrition with th 3-onntivity o G; s Fig. 3() () n (). Similrly, i ny o th Conitions A.(ii) n B.(i) (ii) is not stisi, thn th imrml yl oms srting n hn th ir {, } oms srtion ir o G, gin ontrition with th 3-onntivity o G; s Fig. 3() (), () n (g). Corollry 1. Lt G 3-onnt 1-lnr grh with lnr-mximl 1-lnr ming E(G). Thn no thr rossing g-irs in E(G) shr th sm s g. Proo. Eh rossing g ir inus ithr n ugmnt B- or n ugmnt X- onigurtion. This t long with Lmm 4[A.(i), B.(iii)] yils th orollry. Lmm 5. Lt G 3-onnt 1-lnr grh. Thn thr is lnr-mximl 1- lnr ming E(G ) o surgrh G o G so tht E(G ) ontins t most on ugmnt W-onigurtion in th outr n no othr ugmnt B-onigurtion, n h ugmnt X-onigurtion in E(G ) ontins no vrtx insi its sklton. Proo. Lt E(G) 1-lnr ming o G. W lim tht y norml lnrmximl ugmnttion o E(G) w gt th sir ming o surgrh o G. Not tht u to th g-rrouting this ortion onvrts ny B-onigurtion whos s is not shr with nothr onigurtion into n X-onigurtion; s Fig. 4(). I s g is shr y two B-onigurtions, thy r onvrt into on W - onigurtion n y Lmm 4 this W -onigurtion is on th outr ; s Fig. 4().
() () () Fig. 4. Illustrtion or th roo o Lmm 5. By Corollry 1, s g nnot shr y mor thn two B-onigurtions. Furthrmor this ortion os not rt ny nw B-onigurtion. It lso mks th sklton o ny ugmnt X-onigurtion vrtx-mty; y Lmm 4 s g n shr y t most two ugmnt X-onigurtions rom oosit sis n i it is shr y two ugmnt X-onigurtions, th intrior o th inu imrml yl is mty; s Fig. 4(). Lmm 5 togthr with Proosition 1 imlis th ollowing: Thorm 1. A 3-onnt 1-lnr grh mits stright-lin 1-lnr rwing xt or t most on g in th outr. 4 Gri Drwings In th rvious stion w show tht 3-onnt 1-lnr grh hs strightlin 1-lnr rwing, with th xtion o singl g in th outr. W now strngthn this rsult n show tht thr is stright-lin gri rwing with O(n 2 ) r, whih n onstrut in linr tim rom givn 1-lnr ming. Th lgorithm tks n ming E(G) n omuts norml lnr-mximl ugmnttion. Consir th lnr sklton P(E(G)) or th ming. I thr is n ugmnt W-onigurtion n rossing in th outr, on rossing g on th outr is kt n th othr rossing g is trt srtly. Thus th outr o P(E((G)) is tringl n th innr s r tringls or urngls. Eh urngl oms rom n ugmnt X-onigurtion. It must rwn stritly onvx, suh tht th rossing gs n r-insrt. This is hiv y n xtnsion o th onvx gri rwing lgorithm o Chrok n Knt [8], whih itsl is n xtnsion o th
shiting mtho o Fryssix, Ph n Pollk [15]. Sin th s r t most urngls, w n voi thr ollinr vrtis n th gnrtion to tringl y n xtr unit shit. Not tht our rwing lgorithm hivs n r o (2n 2) (2n 3), whil th gnrl lgorithms or stritly onvx gri rwings [1, 7] ruir lrgr r, sin stritly onvx rwings o n-gons n Ω(n 3 ) r [2]. Brny n Rot giv stritly onvx gri rwing o lnr grh on 14n 14n gri i th s r t most 4-gons, n on 2n 2n gri i, in ition, th outr is tringl. Howvr, thir roh is uit omlx n os not immitly yil ths ouns. It is lso not lr how to us this roh or lnr grhs in our 1-lnr grh stting, in rtiulr whn w hv n unvoil W-onigurtion in th outr. Th lgorithm o Chrok n Knt n in rtiulr th omuttion o nonil omosition rsums 3-onnt lnr grh. Thus th lnr sklton o 3- onnt 1-lnr grh must 3-onnt, whih hols xt or th K 4, whn it is m s n ugmnt X-onigurtion. This rsults rllls th t tht th lnriztion o 3-onnt 1-lnr grh is 3-onnt [14]. Lmm 6. Lt G grh with lnr-mximl 1-lnr ming E(G) suh tht it hs no ugmnt B-onigurtion n h ugmnt X-onigurtion in E(G) hs no vrtx insi its sklton. Thn th lnr sklton P(E(G)) is 3-onnt. W will rov Lmm 6 y showing tht thr is no srtion ir in P(E(G)). First w otin lnr grh H rom G s ollows. Lt (, ) n (, ) ir o rossing gs tht orm n ugmnt X-onigurtion X in Γ. W thn lt th two gs (, ), (, ); vrtx u n th gs (, u), (, u), (, u), (, u) to tringult th. Cll v ross-vrtx n ll this ortion ross-vrtx insrtion on X. W thn otin H rom G y ross-vrtx insrtion on h ugmnt X-onigurtion. Cll H lnriztion o G n not th st o ll th ross-vrtis y U. Thn P(E(G)) = H \U. Bor roving Lmm 6 w onsir svrl rortis o H, th lnriztion o th 1-lnr grh. Lmm 7. Lt G = (V, E) grh with lnr-mximl 1-lnr ming E(G) suh tht E(G) ontins no ugmnt B-onigurtion n h ugmnt X- onigurtion in E(G) ontins no vrtx insi its sklton. Lt H lnriztion o G, whr U is th st o ross-vrtis. Thn th ollowing onitions hol. () H is mximl lnr grh (xt i H is th K 4 in n X-onigurtion) () Eh vrtx o U hs gr 4. () U is n innnt st o H. () Thr is no srting tringl o H ontining ny vrtx rom U. () Thr is no srting 4-yl o H ontining two vrtis rom U. Proo. For onvnin, w ll h vrtx in V U rgulr vrtx. () Sin H is lnr grh, w only show tht h o H is tringl. Eh rossing g ir in Γ inus n ugmnt X-onigurtion whos sklton hs no vrtx in its intrior. Hn h o H ontining rossing vrtx is tringl. Agin, Hong t l. [17] show tht in lnr-mximl 1-lnr ming with no rossing vrtis is tringl. Thus H is mximl lnr grh.
() () Fig. 5. Illustrtion or th roo o Lmm 6. () () Ths two onitions ollow rom th t tht th nighorhoo o h rossing vrtx onsists o xtly our rgulr vrtis tht orm th sklton o th orrsoning ugmnt X-onigurtion. () For ontrition suos vrtx u U rtiits in srting tringl T o H. Sin th nighorhoo o u orms th sklton o th orrsoning ugmnt X-onigurtion X, th othr two vrtis, sy n, in T r rgulr vrtis. Th g (, ) nnot orm s g or X, sin i it i, thn th intrior o th srting tringl T woul ontin in th intrior o th sklton or X n hn woul mty. Assum thror tht n r not onsutiv on th sklton o X. In this s th g (, ) is rossing g in G n hn hs n lt whn onstruting H; s Fig. 5(). () Suos two vrtis u, v U rtiit in srting 4-yl T o H. Du to Conition (), ssum tht T = uv, whr, r rgulr vrtis. I th two vrtis, r jnt in H, ssum without loss o gnrlity tht th g (, ) is rwn insi th intrior o T. Thn th intrior o t lst on o th two tringls u n v is non-mty, inuing srting tringl in H, ontrition with Conition (). W thus ssum tht th two vrtis n r not jnt in H. Thn or oth th ugmnt X-onigurtions X n Y, orrsoning to th two rossing vrtis u n v, th two vrtis u n v r not onsutiv on thir sklton. This imlis tht th rossing g (, ) rtiits in two irnt ugmnt X-onigurtions in Γ, gin ontrition; s Fig. 5(). W r now ry to rov Lmm 6. Proo (Lmm 6). Assum or ontrition tht P(E(G)) is not 3-onnt. Thn thr xists som srtion ir {, } in P(E(G)). Lt H th lnriztion o G, whr U is th st o ross-vrtis. Thn S = U {, } is srting st or H. Tk miniml srting st S S suh tht no ror sust o S is srting st. Sin H is mximl lnr grh (rom Lmm 7()), S orms srting yl [3]. Th 3-onntivity o th mximl lnr grh H imlis S 3. Agin sin S ontins t most two rgulr vrtis, n no two ross-vrtis n jnt in H (Lmm 7()), S < 5. Hn S is srting tringl or srting 4-yl with t most two rgulr vrtis; w gt ontrition with Lmm 7() ().
Finlly, w sri our lgorithm or stright-lin gri rwings. This rwing lgorithm is s on n xtnsion o th lgorithm o Chrok n Knt [8] or omuting onvx rwing o lnr 3-onnt grh. For onvnin w rr to this lgorithm s th CK-lgorithm n w gin with ri ovrviw. Lt G = (V, E) n m 3-onnt grh n lt (u, v) n g on th outr-yl o G. Th CK-lgorithm strts y omuting nonil omosition o G, whih is n orr rtition V 1, V 2,..., V t o V suh tht th ollowing onitions hol: (i) For h k {1, 2,..., t}, th grh G k inu y th vrtis V 1... V k is 2-onnt n its outr-yl C k ontins th g (u, v). (ii) G 1 is yl, V t is singlton {z}, whr z / {u, v} is on th outr-yl o G. (iii) For h k {2,..., t 1} th ollowing onitions hol: I V k is singlton {z}, thn z is on th outr o G k 1 n hs t lst on nighor in G G k. I V k is hin {z 1,..., z l }, h z i hs t lst on nighor in G G k, z 1, z l hv on nighor h on C k 1 n no othr z i hs nighors on G k 1. For h k {1, 2,..., t}, w sy tht th vrtis tht long to V k hv rnk k. W ll vrtx o G k sturt i it hs no nighor in G G k. Th CK-lgorithm strts y rwing th g (u, v) with horizontl lin-sgmnt o unit lngth. Thn or k = 1, 2,..., t, it inrmntlly omlts th rwing o G k. Lt C k 1 = {(u = w 1,..., w,..., w,..., w r = v)} with 1 < r whr w n w r th ltmost n th rightmost nighor o vrtis in V k. Thn th vrtis o V k r l ov th vrtis w,..., w. Assum tht V k = {z 1,..., z l }. Thn z 1 is l on th vrtil lin ontining w i w is sturt in G k ; othrwis it is l on th vrtil lin on unit to th right o w. On th othr hn, z l is l on th ngtiv igonl lin (i.., with 45 slo) ontining w. I v k is singlton thn z = z 1 = z l is l t th intrstion o ths two lins. Othrwis (tr nssry shiting o w n othr vrtis), th vrtis z 1,... z l r l on onsutiv vrtil lins on unit rt rom h othr. In orr to mk sur tht this shiting ortion os not istur lnrity or onvxity, h vrtx v is ssoit with n unr-st U(v) n whnvr v is shit, ll vrtis in U(v) r lso shit long with v. Thus th gs twn vrtis o ny U(v) r in sns rigi. Thorm 2. Givn 1-lnr ming E(G) o 3-onnt grh G, strightlin rwing on th (2n 2) (2n 3) gri n omut in linr tim. Only on g on th outr my ruir on n. Proo. Assum tht E(G) is norml lnr-mximl ming; othrwis w omut on y norml lnr-mximl ugmnttion in linr tim y Lmm 3. Consir th lnr sklton P(E(G)). I thr is no unvoil W-onigurtion on th outr o th mximl lnr ugmnttion, thn th outr-yl o P(E(G)) is tringl. Othrwis w on o th rossing gs in th outr to P(E(G)) to mk th outr-yl tringl. Th othr rossing g is trt srtly. By Lmm 6, P(E(G)) is 3-onnt, its outr is tringl (,, ) n th innr s r tringls or urngls, whr th lttr rsult rom ugmnt X-onigurtions n r in on-to-on orrsonn to irs o rossing gs.
W wish to otin lnr stright-lin gri rwing o P(E(G)) suh tht ll urngls r stritly onvx. Although th CK-lgorithm rws ny 3-onnt lnr grh o n vrtis on gri o siz (n 1) (n 1) with onvx s, th s r not nssrily stritly onvx [8]. Hn w must moiy th lgorithm so tht ll urngls r stritly onvx. Not tht y th ssignmnt o th unr-sts, th CK-lgorithm gurnts tht on is rwn stritly onvx, it woul rmin stritly onvx tr ny susunt shiting o vrtis. For P(E(G)) h V k is ithr singl vrtx or ir with n g, sin th s r t most urngls. I V k is n g (z 1, z 2 ) thn, y th inition o th nonil omosition, xtly on urngl w z 1 z 2 w is orm n y onstrution this is rwn onvx. W thus ssum tht V k ontins singl vrtx, sy v. Lt C k 1 = {(u = w 1,..., w,..., w,..., w r = v)} with 1 < r whr w n w r th ltmost n th rightmost nighors o vrtis in V k. Thn th nw s rt y th insrtion o v r ll rwn stritly onvx unlss thr is som urngl vw 1w w +1 whr < < n w 1, w, w +1 r ollinr in th rwing o G k 1. In this s w must sturt in G k 1 n this ours in th CK-lgorithm only whn th lin ontining w 1, w, w +1 is vrtil lin or ngtiv igonl (with 45 slo). In th ormr s, w 1 shoul hv lso n sturt in G k 1, whih is not ossil sin v is its nighor. It is thus suiint to nsur tht no sturt vrtx o G k is in th ngtiv igonl o oth its lt n right nighors on C k. W o this y th ollowing xtnsion o th CK-lgorithm. Suos v is l ov w with slo 45, w ws l ov its rightmost lowr nighor w with slo 45, n thr is urngl (v, w, w, u) or som vrtx u with highr rnk to l ltr. Thn shit w y on xtr unit to th right whn v or u is l. This imlis n t w n stritly onvx ngl ov w. Th CK-lgorithm strts y ling th irst two vrtis on unit wy n it ruirs unit shit to th right or h ollowing vrtx. On th othr hn, 1-lnr grh hs t most n 2 irs o rossing gs. Hn, thr r g n 3 ugmnt X-onigurtions, h o whih inus urngl in th lnr sklton. Thus th with n hight r n 1 + g, whih is oun y 2n 4. Th vrtis,, o th outr tringl r l t th gri oints (0, 0), (0, n 1 + g), (n 1 + g, 0). I th grh h n unvoil W -onigurtion in th outr, w n ostrossing hs to rw th xtr g (, ), whih inus rossing with th g (, ). Sin is th ltmost lowr nighor o whn is l n is not sturt, is l t (1, j) or som j < n 2 + g. Shit on unit to th right, insrt n t ( 1, n + g), on igonl unit lt ov n rout (, ) vi th n oint. 5 Conlusion n Futur Work W show tht 3-onnt 1-lnr grhs n m on O(n) O(n) intgr gri, so tht gs r rwn s stright-lin sgmnts (xt or t most on g on th outr tht ruirs n). Morovr, th lgorithm is siml n runs in linr tim givn 1-lnr ming. Not tht vn th my ruir xonntil r or givn ix 1-lnr ming,.g., [17]. Rognition o 1-lnr grhs is NP-hr [18]. How hr is th rognition o lnr-mximl 1-lnr grhs?
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