Straight-line Grid Drawings of 3-Connected 1-Planar Graphs

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Corey Eaton
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1 Strt-ln Gr Drwns o 3-Connt 1-Plnr Grs M. Jwrul Alm 1, Frnz J. Brnnur 2, n Stn G. Koourov 1 1 Drtmnt o Comutr Sn, Unvrsty o Arzon, USA {mlm, koourov}@s.rzon.u 2 Unvrsty o Pssu, Pssu, Grmny rnn@normtk.un-ssu. Astrt. A r s 1-lnr t n rwn n t ln su tt s ross t most on. In nrl, 1-lnr rs o not mt strtln rwns. W sow tt vry 3-onnt 1-lnr r s strt-ln rwn on n ntr r o urt sz, wt t xton o snl on t outr tt s on n. T rwn n omut n lnr tm rom ny vn 1-lnr mn o t r. 1 Introuton Sn Eulr s Könsr r rolm tn k to 1736, lnr rs v rov ntrstn rolms n tory n n rt. Fáry, Stn n Wnr rov nnntly tt vry lnr r s strt-ln lnr rwn [16, 24, 28]. Usn t lort tnus o nonl orrn n Snyr rlzrs, ts rsults wr mrov to strt-ln rwn on r o urt sz, n su rwns n omut n lnr tm [9, 23]. T r oun s symtotlly otml, sn t nst trnl rs r lnr rs n rur Ω(n 2 ) r [11]. T rwn lortms wr rn to mrov t r rurmnt or to mt onvx rrsnttons,.., wr nnr s onvx [5, 8, 19] or strtly onvx [1]. Howvr, most rs r nonlnr n rntly, tr v n mny ttmts to stuy lrr lsss o rs. O rtulr ntrst r 1-lnr rs, w n sns r on st yon lnr rs. Ts wr ntrou y Rnl [22] n n ro to olor lnr r n ts ul. Altou t s known tt 3-onnt lnr r n ts ul v strt-ln 1-lnr rwn [27] n vn on r o urt sz [14], lttl s known out nrl 1-lnr rs. It s NP-r to ronz 1-lnr rs [17, 20] n nrl, ltou tr s lnr-tm tstn lortm [12] or mxml 1-lnr rs (.., wr no tonl n wtout voltn 1-lnrty) vn t t rulr orrn o nnt s roun vrtx. A 1-lnr r wt n vrts s t most 4n 8 s [4, 15, 21] n ts ur oun s tt. On t otr n strt-ln rwns o 1-lnr rs my v t most 4n 9 s n ts oun s tt [10]. Hn not ll 1-lnr rs mt strt-ln rwns. Unlk lnr rs, mxml 1-lnr rs n mu srsr wt only 2.64n s [6]. Tomssn [26] rrs to 1-lnr rs s rs wt ross nx 1 n rov tt n m 1-lnr r n turn nto strt-ln rwn n only t xlus B- n W -onurtons; s F. 2. Ts orn onurtons wr

2 () () () () F. 1. () () A 3-onnt 1-lnr r n ts strt-ln r rwn (wt on n n on ), () () notr 3-onnt 1-lnr r n ts strt-ln r rwn. rst sovr y Elton [13] n us y Hon t l. [18], wo sow tt t onurtons n tt n lnr tm t mn s vn. Ty lso rov tt tr s lnr tm lortm to onvrt 1-lnr mn wtout B- n W - onurtons nto strt-ln rwn, ut wtout ouns or t rwn r. In ts r w sttl t strt-ln r rwn rolm or 3-onnt 1- lnr rs. Frst w omut norml orm or n m 1-lnr rs wt no B-onurton n t most on W -onurton on t outr. Tn, tr umntn t r wt s mny lnr s s ossl n tn ltn t rossn s, w n 3-onnt lnr r, w s rwn wt strtly onvx s usn n xtnson o t lortm o Crok n Knt [8]. Fnlly t rs o rossn s r rnsrt nto t onvx s. Ts vs strt-ln rwn on r o urt sz wt t xton o snl on t outr, w my n on n (n ts xton s unvol); s F. 1. In ton, t rwn s otn n lnr tm rom vn 1-lnr mn. 2 Prlmnrs W onsr sml unrt rs G = (V, E) wt n vrts n m s. A rwn o r s mn o G nto t ln su tt t vrts r m to stnt onts n s Jorn r twn ts nonts. A rwn s lnr t Jorn rs o t s o not ross n t s 1-lnr s ross t most on. Not tt rossns twn s nnt to t sm vrtx r not llow. For xml, K 5 n K 6 r 1-lnr rs. An mn o r s lnr (rs. 1-lnr) t mts lnr (rs. 1-lnr) rwn. An mn ss t s, w r toololly onnt rons. T unoun s t outr. A n lnr r s s y yl sun o s on ts ounry (or uvlntly y t yl sun o t nonts o t s). Aornly, 1-lnr mn E(G) ss t s n 1-lnr rwn o G nlun t outr. A 1-lnr mn s wtnss or 1-lnrty. In rtulr, E(G) srs t rs o rossn s n t s wr t s ross. E r o rossn s (, ) n (, ) nus rossn ont. Cll t smnt o n twn t vrtx n t rossn ont l-. E l s mrml, nloous to t s n lnr rwns, n t sns tt no n ross su l- wtout voltn t 1-lnrty o t mn. T non-ross s r ll lnr. A lnrzton G s otn rom E(G) 2

3 () () () F. 2. () An umnt X-onurton, () n umnt B-onurton, () n umnt W -onurton. T rs nu y t sol s r ll n X-onurton (), B-onurton (), n W -onurton (). y usn t rossn onts s rulr vrts n rln rossn y ts two l-s. A 1-lnr mn E(G) n ts lnrzton sr uvlnt mns, n s vn y lst o s n l-s nn t, or uvlntly, y lst o vrts n rossn onts o t s n l s. Elton [13] rs t rolm o ronzn 1-lnr rs wt rtlnr rwns. H solv ts rolm or outr-1-lnr rs (1-lnr rs wt ll vrts on t outr-yl) n roos tr orn onurtons. Tomssn [26] solv Elton s rolm n rtrz t rtlnr 1-lnr mns y t xluson o B- n W-onurtons; s F. 2. Hon t l. [18], otn smlr rtrzton wr t B- n W -onurtons r ll t Bulr n Gu rs. Ty lso sow tt ll ourrns o ts onurtons n omut n lnr tm rom vn 1-lnr mn. Dnton 1. Consr 1-lnr mn E(G): A B-onurton onssts o n (, ) n two s (, ) n (, ) w ross n som ont su tt n l n t ntror o t trnl (,, ). Hr (, ) s ll t s o t onurton. An X-onurton onssts o r (, ) n (, ) o rossn s w os not orm B-onurton. A W-onurton onssts o two rs o s (, ), (, ) n (, ), (, ) w ross n onts n, su tt,,, l n t ntror o t urnl,,,. Hr n t (,, ), rsnt s t s. Osrv tt or ll ts onurtons t s s my ross y notr, wrs t rossn s r mrml; s F 2. Tomssn [26] n Hon t l. [18] rov tt or 1-lnr mn to mt strt-ln rwn, B- n W -onurtons must xlu: Prooston 1. A 1-lnr mn E(G) mts strt-ln rwn wt toololly uvlnt mn n only t os not ontn B- or W -onurton. Aumnt vn 1-lnr mn E(G) y n s mny s to E(G) s ossl so tt G rmns sml r n t nwly s r lnr n E(G). W ll su n mn lnr-mxml mn o G n t orton 3

4 lnr-mxml umntton. (Not tt Hon t l. [18] olor t lnr s o 1- lnr mn s r n ll lnr-mxml umntton r umntton.) T lnr sklton P(E(G))) onssts o t lnr s o lnr-mxml umntton. It s lnr m r, sn ll rs o rossn s r omtt. Not tt t lnr umntton n t lnr sklton r n or n mn, not or r. A r my v rnt mns w v rs to rnt onurtons n umnttons. T noton o lnr-mxml mn s rnt rom t notons o mxml 1-lnr mns n mxml 1-lnr rs, w r su tt t ton o ny volts 1-lnrty (or smlty) [6]. T ollown lm, rovn n mny rlr rs [6, 15, 18, 25, 26], sows tt rossn r o s nus K 4 n lnr-mxml mn, sn mssn s o K 4 n wtout nun nw rossns. Lmm 1. Lt E(G) lnr-mxml 1-lnr mn o r G n lt (, ) n (, ) two rossn s. Tn t our vrts {,,, } nu K 4. By Lmm 1, or lnr-mxml mn X-, B, n W -onurton s umnt y tonl s. Hr w n ts umnt onurtons. Dnton 2. Lt E(G) lnr-mxml 1-lnr mn o r G. An umnt X-onurton onssts o K 4 wt vrts (,,, ) su tt t s (, ) n (, ) ross ns t urnl. An umnt B-onurton onssts o K 4 wt vrts (,,, ) su tt t s (, ) n (, ) ross yon t ounry o t urnl. An umnt W-onurton onssts o two K 4 s (,,, ) n (,,, ) on o w s n n umnt X-onurton n t otr n n umnt B-onurton. For n umnt X- or umnt B-onurton, t s not nun rossn wt otr s n t onurton ns yl, w ll t t sklton. In onurton, t s on t outr-ounry o t m onurton n not nun rossn wt otr s n t onurton r t s s. Usn t rsults o Tomssn [26] n Hon t l. [18], w n now rtrz wn lnr-mxml 1-lnr mn o r mts strt-ln rwn: Lmm 2. Lt E(G) lnr-mxml 1-lnr mn o r G. Tn tr s strt-ln 1-lnr rwn o G wt toololly uvlnt mn s n E(G) n only E(G) os not ontn n umnt B-onurton. Proo. Assum tt E(G) ontns n umnt B-onurton. Tn t must ontn B-onurton n s no strt-ln 1-lnr rwn u to Prooston 1. Convrsly, E(G) s no strt-ln 1-lnr rwn tn y t ontns t lst on B- or W -onurton. Sn Γ s lnr-mxml mn, y Lmm 1 rossn r n E(G) nus K 4. Tus t ott s n F. 2() () must rsnt n ny B- or W - onurton, nun n umnt B-onurton. T norml orm or n m 1-lnr r E(G) s otn y rst n t our lnr s to orm K 4 or r o rossn wl routn tm losly to t rossn s n tn rmovn ol ult s nssry. Su 4

5 n mn o 1-lnr r s norml mn o t. A norml lnrmxml umntton or n m 1-lnr r s otn y rst nn norml orm o t mn n tn y lnr-mxml umntton. Lmm 3. Gvn 1-lnr mn E(G), t norml lnr-mxml umntton o E(G) n omut n lnr tm. Proo. Frst umnt rossn o two s (, ) n (, ) to K 4, su tt t s (, ), (, ), (, ), (, ) r n n s o ult t ormr s rmov. Tn ll umnt X-onurtons r mty n ontn no vrts ns tr skltons. Nxt trnult ll s w o not ontn l-, rossn, or rossn ont. E st n on n lnr tm. 3 Crtrzton o 3-Connt 1-Plnr Grs Hr w rtrz 3-onnt 1-lnr rs y norml mn, wr t rossns r umnt to K 4 s su tt t rsultn umnt X-onurtons v vrtx-mty skltons n tr s no umnt B-onurton xt or t most on umnt W-onurton wt r o rossn s n t outr. Lt E(G) 1-lnr mn o r G. E r o rossn s nus rossn ont n t rossn s n tr l-s r mrml s ty nnot ross y otr s wtout voltn 1-lnrty. An mrml t n E(G) s n ntrnlly-sont sun P = v 1, 1, v 2, 2,..., v n, n, v n+1, wr v 1, v 2,..., v n+1 r (rulr) vrts o G, 1, 2,..., n r rossn onts n E(G) n (v, ), (, v +1 ) or {1, 2,..., n} r l s. I v n+1 = v 0, tn P s n mrml yl. An mrml yl s srtn wn t s vrts ot ns n outs o t, sn ltn ts vrts sonnts G. Lmm 4. Lt G = (V, E) 3-onnt 1-lnr r wt lnr-mxml 1-lnr mn E(G). Tn t ollown ontons ol. A. () Two umnt B-onurtons or two umnt X-onurtons nnot on t sm s o ommon s. () Suos n umnt B-onurton B n n umnt X-onurton X r on t sm s o ommon s (, ). Lt n t rossn onts or X n B, rstvly n lt R(X) n R(B) t rons ns t skltons o X n B. Tn ll vrts o V \ {, } r ns t mrml yl R(X) R(B); otrws ll vrts o V \ {, } r outs t mrml yl. B. () I two umnt B-onurtons r on oost ss o ommon s (, ), wt rossn onts n, rstvly, tn ll t vrts o V \ {, } r ns t mrml yl. () I two umnt X-onurtons r on oost ss o ommon s (, ), wt rossn onts n, rstvly, tn ll t vrts o V \ {, } r outs t mrml yl. () An umnt B-onurton n n umnt X-onurton nnot sr ommon s rom oost ss. Proo. Conton A.() n B.() ol us o ts onurtons nus srtn mrml yl n E(G) wt only two (rulr) vrts rom G, 5

6 () () () () () () () F. 3. Illustrton or t roo o Lmm 4. ontrton wt t 3-onntvty o G; s F. 3() () n (). Smlrly, ny o t Contons A.() n B.() () s not sts, tn t mrml yl oms srtn n n t r {, } oms srton r o G, n ontrton wt t 3-onntvty o G; s F. 3() (), () n (). Corollry 1. Lt G 3-onnt 1-lnr r wt lnr-mxml 1-lnr mn E(G). Tn no tr rossn -rs n E(G) sr t sm s. Proo. E rossn r nus tr n umnt B- or n umnt X- onurton. Ts t lon wt Lmm 4[A.(), B.()] yls t orollry. Lmm 5. Lt G 3-onnt 1-lnr r. Tn tr s lnr-mxml 1-lnr mn E(G ) o sur-r G o G su tt E(G ) ontns t most on umnt W-onurton n no otr umnt B-onurton, n umnt X-onurton n E(G ) ontns no vrtx ns ts sklton. Proo. Lt E(G) 1-lnr mn o G. W lm tt y norml lnrmxml umntton o E(G) w t t sr mn o surr o G. Not tt u to t -rroutn ts orton onvrts ny B-onurton wos s s not sr wt notr onurton nto n X-onurton; s F. 4(). On t otr n s s sr y two B-onurtons, ty r onvrt nto on W -onurton n y Lmm 4 ts W -onurton must on t outr; s F. 4(). By Corollry 1, s nnot sr y mor tn two umnt B onurtons. Furtrmor ts orton os not rt ny nw B-onurtons. It lso mks t sklton o ny umnt X-onurton vrtx-mty, sn y Lmm 4 t sm s n sr y t most two umnt X-onurtons rom t oost s n n s t s sr y two umnt X-onurton, t ntror o t nu mrml yl s mty; s F. 4(). Lmm 5 totr wt Prooston 1 mls t ollown: Torm 1. A 3-onnt 1-lnr r mts strt-ln 1-lnr rwn xt or t most on n t outr. 6

7 () () () F. 4. Illustrton or t roo o Lmm 5. 4 Gr Drwns In t rvous ston w sow tt 3-onnt 1-lnr r s strt-ln 1-lnr rwn, wt t xton o snl n t outr, w oms rom n unvol W-onurton. W now strntn ts rsult n sow tt tr s strt-ln r rwn wt O(n 2 ) r, w n onstrut n lnr tm rom vn 1-lnr mn. T lortm tks n mn E(G) n omuts norml lnr-mxml umntton. Consr t lnr sklton P(E(G)) or t mn. I tr s n umnt W-onurton n rossn n t outr, on rossn on t outr s kt n t otr rossn s trt srtly. Tus t outr o P(E((G)) s trnl n t nnr s r trnls or urnls. E urnl oms rom n umnt X-onurton. It must rwn strtly onvx, su tt t rossn s n r-nsrt. Ts s v y n xtnson o t onvx r rwn lortm o Crok n Knt [8], w tsl s n xtnson o t stn mto o Fryssx, P n Pollk [9]. Sn t s r t most urnls, w n vo tr ollnr vrts n t nrton to trnl y n xtr unt st. Not tt our lortm vs O(n 2 ) r, wl t nrl lortms or strtly onvx r rwns [1, 7] rur lrr r, sn strtly onvx rwns o n-ons n Ω(n 3 ) r [2]. T lortm o Crok n Knt n n rtulr t omutton o nonl omoston rsums 3-onnt lnr r. Tus t lnr sklton o 3- onnt 1-lnr r must 3-onnt, w ols xt or t K 4, wn t s m s n umnt X-onurton. Ts rsults rllls t t tt t lnrzton o 3-onnt 1-lnr r s 3-onnt [15]. Lmm 6. Lt G r wt lnr-mxml 1-lnr mn E(G) su tt t s no umnt B-onurton n umnt X-onurton n E(G) s no vrtx ns ts sklton. Tn t lnr sklton P(E(G)) s 3-onnt. 7

8 W wll rov Lmm 6 y sown tt tr s no srton r n P(E(G)). Frst w otn lnr r H rom G s ollows. Lt (, ) n (, ) r o rossn s tt orm n umnt X-onurton X n Γ. W tn lt t two s (, ), (, ); vrtx u n t s (, u), (, u), (, u), (, u) to trnult t. Cll v ross-vrtx n ll ts orton ross-vrtx nsrton on X. W tn otn H rom G y ross-vrtx nsrton on umnt X-onurton. Cll H lnrzton o G n not t st o ll t ross-vrts y U. Tn P(E(G)) = H \U. Bor rovn Lmm 6 w onsr svrl rorts o H, t lnrzton o t 1-lnr r. Lmm 7. Lt G = (V, E) r wt lnr-mxml 1-lnr mn E(G) su tt E(G) ontns no umnt B-onurton n umnt X- onurton n E(G) ontns no vrtx ns ts sklton. Lt H lnrzton o G, wr U s t st o ross-vrts. Tn t ollown ontons ol. () H s mxml lnr r (xt H s t K 4 n n X-onurton) () E vrtx o U s r 4. () U s n nnnt st o H. () Tr s no srtn trnl o H ontnn ny vrtx rom U. () Tr s no srtn 4-yl o H ontnn two vrts rom U. Proo. For onvnn, w ll vrtx n V U rulr vrtx. () Sn H s lnr r, y nton w only n to sow tt o H s trnl. E rossn r n Γ nus n umnt X-onurton wos sklton s no vrtx n ts ntror. Tror o H ontnn rossn vrtx s trnl. On t otr n, Hon t l. [18] sow tt n ny lnr-mxml 1-lnr mn ontnn no rossn vrts s trnl. Tus H s mxml lnr r. () () Ts two ontons ollow rom t t tt t noroo o rossn vrtx onssts o xtly our rulr vrts tt orm t sklton o t orrsonn umnt X-onurton. () For ontrton suos vrtx u U rtts n srtn trnl T o H. Sn t noroo o u orms t sklton o t orrsonn umnt X-onurton X, t otr two vrts, sy n, n T r rulr vrts. T (, ) nnot orm s or X, sn t, tn t ntror o t srtn trnl T woul ontn n t ntror o t sklton or X n n woul mty. Assum tror tt n r not onsutv on t sklton o X. In ts s t (, ) s rossn n G n n s n lt wn onstrutn H; s F. 5(). () Suos two vrts u, v U rtt n srtn 4-yl o H. Du to Conton (), ssum wtout loss o nrlty tt t srtn 4-yl s T = uv, wr, r rulr vrts. Assum rst tt t two vrts, r nt n H n lso ssum wtout loss o nrlty tt t (, ) s rwn ns t ntror o T. Ts mns tt t ntror o t lst on o t two trnls u n v s non-mty n n t lst on o ts two trnls orms srtn trnl n H, ontrton wt Conton (). W 8

9 () () F. 5. Illustrton or t roo o Lmm 6. tus ssum tt t two vrts n r not nt n H. Tn or ot t umnt X-onurtons X n Y, orrsonn to t two rossn vrts u n v, t two vrts u n v r not onsutv on tr sklton. Ts mls tt t rossn (, ) rtts n two rnt umnt X- onurtons n Γ, n ontrton; s F. 5(). W r now ry to rov Lmm 6 Proo (Lmm 6). Assum or t uros o otnn ontrton tt P(E(G)) s not 3-onnt. Tn tr xsts som srton r {, } n P(E(G)). Lt H t lnrzton o G, wr U s t st o ross-vrts. Tn S = U {, } s srtn st or H. Tk mnml srtn st S S su tt no ror sust o S s srtn st or H. Sn H s mxml lnr r (rom Lmm 7()), S must orm srtn yl [3]. As H s mxml lnr r t must 3- onnt, w mls tt S 3. On t otr n, sn S ontns t most two rulr vrts, n no two ross-vrts n nt n H (Lmm 7()), S < 5. Hn S s tr srtn trnl or srtn 4-yl n H ontnn t most two rulr vrts; w t ontrton wt Lmm 7() (). Fnlly, w sr our lortm or strt-ln r rwns. Ts rwn lortm s s on n xtnson o t lortm o Crok n Knt [8] or omutn onvx rwn o lnr 3-onnt r. For onvnn w rr to ts lortm s t CK-lortm n w n wt r ovrvw. Lt G = (V, E) n m 3-onnt r n lt (u, v) n on t outr-yl o G. T CK-lortm strts y omutn nonl omoston o G, w s n orr rtton V 1, V 2,..., V t o V su tt t ollown ontons ol: () For k {1, 2,..., t}, t r G k nu y t vrts V 1... V k s 2-onnt n ts outr-yl C k ontns t (u, v). () G 1 s yl, V t s snlton {z}, wr z / {u, v} s on t outr-yl o G. () For k {2,..., t 1} t ollown ontons ol: I V k s snlton {z}, tn z s on t outr o G k 1 n s t lst on nor n G G k. I V k s n {z 1,..., z l }, z s t lst on nor n G G k, z 1, z l v on nor on C k 1 n no otr z s nors on G k 1. 9

10 For k {1, 2,..., t}, t vrts tt lon to V k v rnk k. W ll vrtx o G k sturt t s no nor n G G k. T CK-lortm strts y rwn t (u, v) wt orzontl ln-smnt o unt lnt. Tn or k = 1, 2,..., t, t nrmntlly omlts t rwn o G k. Lt C k 1 = {(u = w 1,..., w,..., w,..., w r = v)} wt 1 < r su tt w n w r t ltmost n t rtmost nor o vrts n V k. Tn t vrts o V k r l ov t vrts w,..., w. Assum tt V k = {z 1,..., z l }. Tn z 1 s l on t vrtl ln ontnn w w s sturt n G k ; otrws t s l on t vrtl ln on unt to t rt o w. On t otr n, z l s l on t ntv onl ln (.., wt 45 slo) ontnn w. I v k s snlton tn z = z 1 = z l s l t t ntrston o ts two lns. Otrws (tr nssry stn o w n otr vrts), t vrts z 1,... z l r l on onsutv vrtl lns on unt rt rom otr. In orr to mk sur tt ts stn orton os not stur lnrty or onvxty, vrtx v s ssot wt n unr-st U(v) n wnvr v s st, ll vrts n U(v) r lso st lon wt v. Tus t s twn vrts o ny U(v) r n sns r. Torm 2. Gvn 1-lnr mn E(G) o 3-onnt r G, strtln rwn on t (2n 2) (2n 3) r n omut n lnr tm. Only on on t outr my rur on n. Proo. Assum tt E(G) s norml lnr-mxml mn; otrws w omut on y norml lnr-mxml umntton n lnr tm y Lmm 3. Consr t lnr sklton P(E(G)). I tr s no unvol W-onurton on t outr o t mxml lnr umntton, tn t outr-yl o P(E(G)) s trnl. Otrws w on o t rossn s n t outr to P(E(G)) to mk t outr-yl trnl. T otr rossn s trt srtly. By Lmm 6, P(E(G)) s 3-onnt, ts outr s trnl (,, ) n t nnr s r trnls or urnls, wr t lttr rsult rom umnt X-onurtons n r n on-to-on orrsonn to rs o rossn s. W ws to otn lnr strt-ln r rwn o P(E(G)) su tt ll urnls r strtly onvx. Altou t CK-lortm rws ny 3-onnt lnr r o n vrts on r o sz (n 1) (n 1) wt onvx s, t s r not nssrly strtly onvx [8]. Hn w must moy t lortm so tt ll urnls r strtly onvx. Not tt y t ssnmnt o t unr-sts, t CK-lortm urnts tt on s rwn strtly onvx, t woul rmn strtly onvx tr ny susunt stn o vrts. For P(E(G)) V k s tr snl vrtx or r wt n, sn t s r t most urnls. I V k s n (z 1, z 2 ) tn, y t nton o t nonl omoston, xtly on urnl w z 1 z 2 w s orm n y onstruton ts s rwn onvx. W tus ssum tt V k ontns snl vrtx, sy v. Lt C k 1 = {(u = w 1,..., w,..., w,..., w r = v)} wt 1 < r su tt w n w r t ltmost n t rtmost nor o vrts n V k. Tn t nw s rt y t nsrton o v r ll rwn strtly onvx unlss tr s som urnl vw 1w w +1 wr < < n w 1, w, w +1 r ollnr n t rwn o G k 1. Not tt n ts s t vrtx w must sturt n G k 1. 10

11 Ts s my our n t CK-lortm only wn t ln ontnn w 1, w, w +1 s tr vrtl ln or ntv onl (wt 45 slo). In t ormr s, w 1 soul v lso n sturt n G k 1, w s not ossl sn v s ts nor. Tus t s sunt to mk sur tt no sturt vrtx o G k s n t ntv onl o ot ts lt n rt nor on C k. W o ts y t ollown xtnson o t CK-lortm. Suos tt v s l ov w wt slo 45 n w ws l ov ts rtmost lowr nor w wt slo 45, n tr s t urnl (v, w, w, u) or som vrtx u wt r rnk (.., w wll l ltr). Tn st w y on xtr unt to t rt wn v or u s l. Ts mls n t w n sts strtly onvx nl ov w. T CK-lortm strts y ln t rst two vrts on unt wy n t rurs unt st to t rt or ollown vrtx. On t otr n, 1-lnr r s t most n 2 rs o rossn s. Hn, tr r n 3 umnt X-onurtons, o w nus urnl n t lnr sklton. Tus t wt n t r n 1 +, w s oun y 2n 4. T vrts,, o t outr trnl r l t t r onts (0, 0), (0, n 1 + ), (n 1 +, 0). In s t ornl r n unvol W -onurton n t outr, w n ost-rossn s to to t rwn t xtr (, ), w nus rossn n t outr wt t (, ). Sn s t ltmost lowr nor o wn s l n s not sturt, s l n t rst olumn t (1, ) or som < n 2 +. St on unt to t rt, nsrt n ont t ( 1, n + ) ust on onl unt lt ov n rout t (, ) v t n ont. Fur 6 n t nx llustrts t orton o t lortm or omutn strt-ln rwn o 3-onnt 1-lnr r on (2n 2) (2n 3) r. 5 Conluson n Futur Work W v sown tt 3-onnt 1-lnr rs n m on t O(n) O(n) ntr r, so tt s r rwn s strt-ln smnts (xt or t most on on t outr tt rurs n). Morovr, t lortm s sml n runs n lnr tm. Som 1-lnr m rs my rur xonntl r; s Hon t l. [18]. As w v sown, ts nnot n wt 3-onnt 1-lnr rs. It s not lr wtr tr xst onnt 1-lnr rs or w ny strtln 1-lnr rwn rurs xonntl r. Ronton o 1-lnr rs s NPr [20]. How r s t ronton o lnr-mxml 1-lnr rs? Rrns 1. I. Bárány n G. Rot. Strtly onvx rwns o lnr rs. Doumnt Mtmt, 11: , I. Bárány n N. Tokus. T mnmum r o onvx ltt n-ons. Comntor, 24(2): , I. Byrs. On k-t mltonn mxml lnr rs. Dsrt Mtmts, 40(1): , R. Bonk, H. Sumr, n K. Wnr. Bmrkunn zu nm Ssrnrolm von G. Rnl. A. us m Mt. Smnr r Unv. Hmur, 53:41 52,

12 5. N. Bonon, S. Flsnr, n M. Mos. Convx rwns o 3-onnt ln rs. Alortm, 47(4): , F.-J. Brnnur, D. Estn, A. Glßnr, M. T. Goor, K. Hnur, n J. Rslur. On t nsty o mxml 1-lnr rs. In Gr Drwn (GD 12), volum 7704 o Ltur Nots n Comutr Sn, s Srnr, M. Crok, M. T. Goor, n R. Tmss. Convx rwns o rs n two n tr mnsons. In Symosum on Comuttonl Gomtry (SoCG 96), s , M. Crok n G. Knt. Convx r rwns o 3-onnt lnr rs. Intrntonl Journl o Comuttonl Gomtry n Altons, 7(3): , H. Fryssx, J. P, n R. Pollk. How to rw lnr r on r. Comntor, 10(1):41 51, W. Dmo. Dnsty o strt-ln 1-lnr r rwns. Inormton Prossn Lttr, 113(7): , D. Dolv, T. Lton, n H. Trky. Plnr mn o lnr rs. In Avns n Comutn Rsr, s , P. Es, S.-H. Hon, N. Kto, G. Lott, P. Swtzr, n Y. Suzuk. Tstn mxml 1-lnrty o rs wt rotton systm n lnr tm. In Gr Drwn (GD 12), s , R. B. Elton. Rtlnr rwns o rs. Utlts Mt., 29: , C. Ertn n S. G. Koourov. Smultnous mn o lnr r n ts ul on t r. Tory o Comutn Systms, 38(3): , I. Fr n T. Mrs. T strutur o 1-lnr rs. Dsrt Mtmts, 307(7 8): , I. Fáry. On strt lns rrsntton o lnr rs. At Sntrum Mtmtrum, 11: , A. Grorv n H. L. Bolnr. Alortms or rs ml wt w rossns r. Alortm, 49(1):1 11, S.-H. Hon, P. Es, G. Lott, n S.-H. Poon. Fáry s torm or 1-lnr rs. In Intrntonl Conrn on Comutn n Comntors (COCOON 12), volum 7434 o Ltur Nots n Comutr Sn, s Srnr, G. Knt. Drwn lnr rs usn t nonl orrn. Alortm, 16:4 32, V. P. Korzk n B. Mor. Mnml ostrutons or 1-mmrsons n rnss o 1- lnrty tstn. Journl o Gr Tory, 72(1):30 71, J. P n G. Tót. Grs rwn wt w rossns r. Comntor, 17: , G. Rnl. En Ssrnrolm u r Kul. A. us m Mt. Smnr r Unv. Hmur, 29: , W. Snyr. Emn lnr rs on t r. In Symosum on Dsrt Alortms (SODA 90), s , S. K. Stn. Convx ms. Amrn Mtmtl Soty, 2(3): , Y. Suzuk. R-mns o mxmum 1-lnr rs. SIAM Journl o Dsrt Mtmts, 24(4): , C. Tomssn. Rtlnr rwns o rs. Journl o Gr Tory, 12(3): , W. T. Tutt. How to rw r. Pro. Lonon Mt. Soty, 13(52): , K. Wnr. Bmrkunn zum Vrrnrolm. Jrsrt r Dutsn Mtmtkr-Vrnun, 46:26 32,

13 Anx Illustrton o t Alortm or 3-Connt 1-lnr Gr () () () () () () F. 6. () A 3-onnt 1-lnr r G, () norml mn or G, () lnr-mxml norml mn E (G), () lnr sklton P(E (G)) omut rom E (G) y ltn t rossn s xt on rossn on t outr, () strt-ln strtly-onvx r rwn Γ o P(E (G)) usn n xtnson o t lortm n [8], n () r rwn Γ o G wt strt-ln s xt or on wt on n. 13

The University of Sydney MATH 2009

The University of Sydney MATH 2009 T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n