CCCG 011, Toronto ON, August 10 1, 011 Outrplanar graphs and Dlaunay triangulations Ashraful Alam Igor Rivin Ilana Strinu Abstract Ovr 0 yars ago, Dillncourt [1] showd that all outrplanar graphs can b ralizd as Dlaunay triangulations of points in convx position. His proof is lmntary, constructiv and lads to a simpl algorithm for obtaining a concrt Dlaunay ralization. In this not, w provid two nw, altrnat, also quit lmntary proofs. 1 Introduction Th Dlaunay triangulation of a point st in convx position is, combinatorially, an outrplanar graph. Dillncourt [1] has shown, constructivly, that th othr dirction is also tru: any graph which ariss from a triangulation of th intrior of a simpl polygon can b ralizd as a Dlaunay triangulation. Dillncourt s proof uss a simpl and natural critrion on th angls of triangls in a Dlaunay triangulation, and givs an O(n ) tim incrmntal algorithm to calculat thir angls and infr a ralization. Lambrt [] adaptd this mthod into a linar tim algorithm, whos implmntation in Java is availabl at http://www.cs.unsw.du.au/ lambrt/java/raliz/ Th gnral qustion, of charactrizing and rconstructing arbitrary Dlaunay triangulations (in twoor highr dimnsions), is substantially mor difficult. A closly rlatd problm, going back to Stinr (s Grünbaum [6], pag 8), asks for a charactrization of th graphs of inscribabl or circumscribabl polyhdra: thos whos vrtics li on a sphr, rsp. whos facs ar tangnt to a sphr. Such graphs ar said to b of inscribabl or circumscribabl typ. Th bst rsult to dat is du to Rivin [], who provd ncssary and sufficint conditions for a polyhdral graph to b of inscribabl or circumscribabl typ. Dillncourt and Smith [3] linkd inscribability of a graph to its ralizability as a Dlaunay triangulation, and gav a critrion rlating Hamiltonicity to inscribability. Dpartmnt of Computr Scinc, Univrsity of Massachustts Amhrst, MA, ashraful@cs.umass.du. Rsarch supportd by NSF CCF-1016988 and DARPA HR0011-09-0003 grants. Dpartmnt of Mathmatics, Tmpl Univrsity, Philadlphia, rivin@math.tmpl.du Dpartmnt of Computr Scinc, Smith Collg, Northampton, MA, strinu@smith.du. Rsarch supportd by NSF CCF- 1016988 and DARPA HR0011-09-0003 grants. In this papr, w prsnt two nw simpl and lmntary proofs of th Dlaunay ralizability of outrplanar graphs. W ar not awar of thm prviously apparing in th litratur. Th first on is an asy consqunc of Dillncourt and Smith s [3] critrion rlating Hamiltonianicity and inscribability. Th scond on, which occupis most of this not, uss Rivin s [] inscribability critrion and constructs an xplicit witnss of this inscribability, in th form of crtain wights assignd to th dgs of th graph. Prliminaris A graph G = (V, E) is a collction of vrtics V = {1,, n} and dgs E, whr an dg = ij E is a pair of vrtics i, j V. A graph is planar if it can b drawn in th plan in a way such that no two dgs cross, xcpt prhaps at ndpoints. A planar drawing of a planar graph is calld a plan graph. It subdivids th plan into rgions calld facs. A plan graph has on unboundd fac, calld th outr fac. A plan graph is dnotd by its vrtics, dgs, facs and th outr fac: G = (V, E, F, f). A plan graph whr all vrtics li on th outr fac is calld an outrplanar graph. A stllatd outrplanar graph is obtaind from an outrplanar graph by adding on vrtx and conncting it to all th original vrtics. W call this th stllating vrtx, and all dgs manating from this vrtx th stllating dgs. Two paths btwn two vrtics ar indpndnt if thy do not shar any vrtics xcpt th nd-points. A graph is connctd if thr is a path btwn any two vrtics and it is k-connctd if thr ar k indpndnt paths btwn any two vrtics. A cutst of a graph is th minimal st of dgs whos rmoval maks th graph disconnctd. A cutst is cotrminous if all th dgs manat from a singl point. A graph is polyhdral if it is planar and 3-connctd. In this cas, th facs of a plan ralization ar uniquly dtrmind up to th choic of th outr fac. By Stinitz thorm (s Grünbaum [6]), any polyhdral graph can b ralizd as a convx polyhdron. A polyhdral graph is inscribabl if its corrsponding convx polyhdron is combinatorially quivalnt to th dgs and vrtics of th convx hull of a st of noncoplanar points on th surfac of th sphr. Givn a st P of points in th Euclidan plan, a triangulation of ths points is a planar graph whr all
3 d Canadian Confrnc on Computational Gomtry, 011 facs, with th possibl xcption of th outr fac, ar triangls. A Dlaunay triangulation of P is a triangulation whr th circumscircl of any triangl dos not contain any othr points of P. Our rsult W giv two nw proofs of Dillncourt s thorm: Thorm 1 Any outrplanar graph can b ralizd as a Dlaunay triangulation. Th first proof Th first proof that an outrplanar graph can b ralizd as a Dlaunay triangulation rlis on two lgant rsults du to Dillncourt and Smith [3] and to Rivin [, 5]. Thy rlat inscribability, ralization as Dlauny triangulation and Hamiltonicity. A Hamiltonian cycl in a graph is a simpl spanning cycl. Any graph that has a Hamiltonian cycl is calld Hamiltonian. A graph is 1-Hamiltonian if rmoving any vrtx from th graph maks it Hamiltonian. Dillncourt and Smith [3] provd: Thorm A 1-Hamiltonian planar graph is of inscribabl typ. Nxt, w nd this rsult of Rivin [, 5]. Thorm 3 A plan graph G = (V, E, F, f), with f as th unboundd fac, is ralizabl as a Dlaunay triangulation for som point st P if and only if th graph G obtaind from G by stllating f is of inscribabl typ. To complt our first proof, w just nd to show that: Lmma A stllatd outrplanar graph G is 1- Hamiltonian. Proof. Lt {1,,, n} b th vrtics of th undrlying outrplanar graph G of G, in countrclockwis ordr on th outr fac, with modulo n indics, and lt s b th stllating vrtx. If w rmov a vrtx i, 1 i n, thn w find a Hamiltonian cycl i + 1, i +,, i 1, s, i + 1. If w rmov vrtx s, w gt th original outrplanar graph G which is Hamiltonian. In th nxt sction, w prsnt our main rsult, which is a mor tchnical proof for th sam rsult. It is basd on a vry gnral critrion of Rivin, and has th advantag of illustrating spcific proprtis (bsids Hamiltonicity) of Dlaunay triangulation ralizations for outrplanar graphs. 3 Th main proof Rivin [, 5] gav this vry gnral critrion for th Stinr s problm: Thorm 5 A planar graph G = (V, E) is of inscribabl typ if and only if it satisfis th following conditions: 1. G is a 3-connctd planar graph.. A st of wights W can b assignd to th dgs of G such that: (a) For ach dg, 0 < w() 1/. (b) For ach vrtx v, th sum of all wights of dgs incidnt to v is 1. (c) For ach non-cotrminous cutst C E, th sum of all th wights of dgs of C must xcd 1. Combining this with Thorm 3, w hav to prov that if G is a stllatd outrplanar graph, thn a wight assignmnt as in Thorm 5 xists. But first, lt us vrify that any stllatd outrplanar graph is 3-connctd. Th following lmma is straightforward: Lmma 6 Any outrplanar graph is -connctd. Proof. In an outrplanar graph, all th vrtics li on th unboundd fac f. If w labl th vrtics as 1,,, n in th ordr in which thy appar on th outr fac, thr ar two indpndnt paths btwn any pair of vrtics i and j: on from i, i + 1,, j and anothr is i, i 1,, j. This lads immdiatly to th vrification of th first condition in Thorm 5: Lmma 7 Any stllatd outrplanar graph G is planar and 3-connctd. Proof. Planarity is straightforward, sinc G was obtaind from an outrplanar graph G by stllating (with a nw vrtx s) its unboundd fac. W show now that thr xists thr indpndnt paths btwn any two vrtics i and j of G. Lt i and j b two vrtics of th outrplanar graph G. Lmma 6 showd that thr ar two indpndnt paths btwn i and j. A third indpndnt path is i, s, j whr s is th stllating vrtx. To complt th proof w show th xistnc of thr indpndnt paths btwn s and any othr vrtx i of G: thy ar (s, i), (s, i 1, i) and (s, i + 1, i), whr indx arithmtic is don modulo n in th rang 1,, n. Notic that w implicitly assum that G has at last thr vrtics, othrwis th thorm is trivial.
CCCG 011, Toronto ON, August 10 1, 011 In th rst of th papr, w dscrib a wight assignmnt for G which satisfis Rivin s critria. In sction 3.1 w giv an inductiv schm to comput th wights, and prov that thy satisfy th first two proprtis (a and b) in Thorm 5. Th proof is compltd in sction 3., whr w vrify th third proprty (c) in Thorm 5. 3.1 Wight assignmnt Instad of assigning wights on th dgs of a stllatd outrplanar graph G, w will assign thm on th dgs of th dual graph G D of G, so first w look closr at th structur of th dual of a stllatd outrplanar graph. Th duals of th stllating dgs of G form a cycl, and th rmaining ons form a tr (dnotd by T D ) whos lavs li on th cycl (s Fig 1). Furthrmor, th tr is partitiond into a path whos vrtics ar not lavs (th backbon) and dgs incidnt to th tr lavs, calld laf-dgs. W thus partition th dgs of G into thr classs, colord blu (cycl), grn (backbon) and black (laf dgs). Primal and dual dgs gt th sam color, s Fig 1(c). On nd of th laf dg is connctd to a backbon dg and anothr nd is connctd to a cycl dg. Th dgs of a fac of G D ar duals of dgs incidnt to a vrtx in G. If thr ar n cycl dgs, thr will b n facs in G D. Lt ths facs b f 1, f,, f n in countr-clockwis ordr. Clarly, it suffics to mak sur that th sum of all wights of cycl dgs and sum of all wights of th dgs of ach fac of G D ar sparatly qual to 1. In addition, w hav to mak sur that th rmaining two conditions a and b of Thorm 5 ar also satisfid. Th wight assignmnt is carrid out in two stps, th contraction stp and th xpansion stp. During contraction, all th backbon dgs ar contractd to obtain a vry spcific typ of dual graph, on which a simpl wight assignmnt is possibl. Th dgs ar thn xpandd back, and adjustmnts to th initial wights ar locally prformd, whil maintaining Rivin s conditions. Contraction: In this stp w contract all th backbon dgs of T D. Thn all th cycl dgs and laf dgs of G D rmain unchangd, but all th facs bcom triangular (s Fig ). Nxt, w assign a wight of 1/n to ach cycl dg. Hr n is th numbr of cycl dgs of G D. Nxt w assign ach laf dg (on of th rmaining two dgs of a triangular fac) a wight of n 1 n. This wight assignmnt satisfis Rivin s conditions a and b for n > 1. (a) (b) Figur : (a)a dual of a stllatd outrplanar graph. Bold dgs ar backbon dgs. (b) Dual graph aftr contraction of backbon dgs. (c) Expansion of a singl backbon dg. Expansion: (a) (b) Figur 1: (a) An outrplanar graph (b) A stllatd outrplanar graph obtaind from (a). Blu dgs ar stllating dgs and blu vrtx is th stllating vrtx. (c) Dual graph of th stllatd outrplanar graph. Dual dgs ar colord according to thir primal dgs. Dark blu, grn and black dgs ar cycl, backbon and laf dgs rspctivly. (c) Now w incrmntally xpand back th backbon dgs of T D. Considr a backbon dg b which is shard by fac f i and fac f j. W assign a wight of 0 < ɛ < 1/ to b, for som positiv ɛ to b dtrmind latr. This crats an imbalanc into th sum of wights for th dgs of facs f i and f j. W rmov this imbalanc by subtracting ɛ from th cycl dgs of f i and f j, and subtracting ɛ from ach of th two laf dgs of f i and f j, rspctivly. Although this rstors th balanc of wights for facs f i and f j, it crats an imbalanc for facs f i 1, f i+1, f j 1, f j+1 and cycl dgs of G D. To fully balanc th wights, w add ɛ to th cycl dgs of ths four facs. This assignmnt of wights mts Rivin s conditions. Th procss is rpatd for ach xpandd backbon dg. S Figur 3.
3 d Canadian Confrnc on Computational Gomtry, 011 1/n / f i 1 unaffctd facs f j+1 xpandd fac xpandd fac f i f j f f j 1 i+1 unaffctd facs 1/n / Th wight assignmnt schm lads to a maximum possibl wight of 1/n for ach dg. Whn n >, this is always smallr than 1/. Whn w xpand a backbon dg, w subtract a valu from som of th dgs of G D, and always add ɛ to two cycl dgs of th adjacnt clls. Th maximum amount that w subtract is ɛ/ from a cycl dg of G D. Thrfor, th only cas whn th wight on any dg bcoms ngativ is whn w subtract mor than th initial valu of th dg. +/ f j+1 1/n / 1/n / xpandd fac xpandd fac f i f j f j 1 f i+1 unaffctd facs xpandd fac f i +/ f j+1 f i+1 xpandd fac f j +/ 1/n / 1/n / Figur 3: Thr possibl cass whn a backbon dg is xpandd. Expandd fac is shard by (a) four distinct facs, (b) two distinct facs and on common fac and (c) two common facs. Whn a backbon dg is xpandd, only wights of ths s hav to b adjustd; wights of othr facs rmain unchangd. Hr w(l) = n 1 n Figur : A stllatd outrplanar graph and its dual (shown in rd dgs) whr th backbon is a singl vrtx. An xtrm cas occurs whn, for ach xpansion of a backbon dg, ɛ/ is subtractd from th sam dg. This is only possibl whn our original outrplanar graph G has all chords manating from a singl vrtx. Considr a fac in G D corrsponding to such vrtx in G. For ach xpansion of th backbon dg, th wight of th cycl dg of that fac is dcrmntd by ɛ/. Similarly, for ach xpansion, th wight of ach of th two cycl dgs is incrasd by ɛ. It rmains to prov that th wight of ach dg of G D satisfis condition a of Thorm 5, if ɛ lis within a crtain rang. Lmma 8 Th wight of ach dg lis within th rang 0 < w() 1/. Proof. W prov this first for th xtrm cas whr a fac f of G D shars all th backbon dg, as in Fig. First w show th uppr bound of w(). Each tim a backbon dg is xpandd, th wight of ach of th two cycl dgs which ar th dgs of two adjacnt facs of f is incrasd by ɛ. Thrfor, ach such dg taks xtra at most ɛ (k + 1) from th wights, whr k is th numbr of backbon dgs in G D or chords in G. In an outrplanar graph, k is xactly n 3. To maintain th uppr bound of w(), w nd 1 n + ɛ(n ) 1 or ɛ n. Now w prov th lowr bound of w(). Each tim ɛ a backbon dg is xpandd, and ɛ ar subtractd from th cycl and laf dgs of th fac f rspctivly. Thrfor, it suffics to show that th final wight of
CCCG 011, Toronto ON, August 10 1, 011 cycl dg rmains positiv. Aftr adjusting wights for all backbon dgs, th final wight of a boundary dg is w() = 1 n (n 3)ɛ, as thr ar (n 3) chords in th outrplanar graph. Sinc th bas wight 1/n is always lss than 1/, w() < 1/. To mak w() positiv, w hav to choos ɛ such that (n 3)ɛ < 1 n or ɛ < n(n 3). This complts th proof, with wights assignmnts w() lying within th rquird rang whn ɛ < min{ n(n 3), n }. 3. Proof of non-cotrminous cutst condition A non-cotrminous cutst is a cutst whr all th dgs of th cutst do not manat from a singl vrtx. A non-cotrminous cutst, lik any othr cutst, divids a connctd graph into two componnts, whr ach componnt consists of at last two vrtics. Sinc ach vrtx in th primal graph is rprsntd by a fac in th dual, th non-cotrminous cutst in th primal is rprsntd by a non-facial cycl in th dual. To prov th non-cotrminous condition c of Thorm 5, w show now that for any non-facial cycl in th dual, th sum of wights of th dgs of th cycl is strictly gratr than 1. For simplicity, lt us considr a non-cotrminous cutst whr th non-facial cycl contains two adjacnt facs in th dual, as in Fig 5. Lt b th dg shard by this two facs. According to condition b, th sum of th wights of th dgs of ach of ths two facs is 1. Thrfor, th sum of th wights of dgs of th non-facial cycl is w(). Sinc th wight of any dg is lss than 1, th wight of th cutst is strictly gratr than 1. It is important to not that this cutst divids th primal graph into two componnts whr on componnt has only two vrtics joind by an dg. Th two facs of th non-facial cycl rprsnts ths two vrtics in th dual and dg is th dual of th conncting dg in th primal. facs f 1, f,, f n. An dg x is calld an xtra dg, if it is shard by two facs f i and f j, whr 1 i j n. Dnot by k th xtra dgs in C and by w max th largst wight possibl on any of ths k dgs (which is ssntially lss than 1 ). In ordr to satisfy th condition c, w nd n kw max > 1. W nd an uppr bound of w max (lowr bound is trivially gratr than 0). In ordr to find that, w nd an uppr bound of k too. Rcall that th xtra dgs in any non-facial cycl rprsnt th dgs of on of th two componnts. Sinc a non-cotrminous cycl divids th primal graph into xactly two componnts, on of th componnts has at last as many vrtics than th othr on. Thrfor, th numbr of vrtics of th smallr componnt is v n and th numbr of dgs in that componnt satisfy v 3 = n 3. Thrfor th uppr bound of k is n 3. Substituting this valu in th quation abov, w gt w max < n 1 (n 3), whr w max is clarly lss than 1. Finally, w nd to convrt w max in trms of ɛ. Initially, in th contractd form, th cycl dgs ar assignd 1 n 1 n ach and laf dgs ar assignd n ach. Both of ths initial wights ar within th bound of w max. Whnvr a backbon dg of fac f i is xpandd, th wights of cycl and laf dgs of f i ar dcrasd. Th only dgs whos wights ar incrasd ar th cycl dgs of fac f i 1 and f i+1. Hnc it is possibl for ths dgs to violat only th w max condition. W gt th bound of w max basd on ths dgs. Th maximum wights of such cycl dgs ar ncountrd whn th outrplanar graph is similar to th xtrm cas statd in lmma 8. In that cas, only on fac f i shars all th backbon dgs and th cycl dgs of facs f i 1 and f i+1 incur maximum additional wights. Sinc thr can b at most n 3 possibl chords in th primal or backbon dgs in th dual, th wight of ach of ths two cycl dg is 1 n + (n 3)ɛ. Thrfor, th wights of ths dgs has to satisfy th condition that 1 n + (n 3)ɛ w max or 1 n + (n 3)ɛ < n 1 (n 3). Solving this quation, w gt ɛ < (n 3n+6) n(n 3). To satisfy both conditions a and c, w will choos an ɛ such that 0 < ɛ < min{ n(n 3), n, (n 3n+6) n(n 3) } This concluds our main proof. Acknowldgmnt This work was initiatd at th 5th Bllairs Wintr Workshop on Computational Gomtry hld January 1 January 7, 010. Figur 5: Th dottd lins show th non-facial cycl in th dual graph. Edg is th xtra dg in th cycl. Now considr a non-facial cycl C which contains n Rfrncs [1] M. B. Dillncourt Ralizability of Dlaunay triangulations. Information Procssing Lttrs, 33(6):83-87, 1990
3 d Canadian Confrnc on Computational Gomtry, 011 [] T. Lambrt An optimal algorithm for ralizing a Dlaunay triangulation Information Procssing Lttrs, 6:5-50, 1997. [3] M. B. Dillncourt and W. D. Smith Graph-thortical conditions for inscribability and Dlaunay ralizability. Discrt Mathmatics, 161(1-3):63 77, Dcmbr, 1996. [] I. Rivin Euclidan Structurs on Simplicial Surfacs and Hyprbolic Volum Th Annals of Mathmatics, 139(3):533 580, May, 199. [5] I. Rivin A charactrization of idal polyhdra in hyprbolic 3-spac. Th Annals of Mathmatics,13(1):51-70, January, 1996. [6] B. Grunbaum Convx Polytops. Wily Intrscinc, 1967.