Codes on Planar Graphs

Transcription

1 Cods on Planar Graphs Srimathy Srinivasan, Andrw Thangaraj Dpartmnt of Eltrial Enginring Indian Institut of Thnology, Madras arxiv: v [s.it] 5 May 009 Astrat Cods dfind on graphs and thir proprtis hav n sujts of intns rnt rsarh. On th pratial sid, onstrutions for apaity-approahing ods ar graphial. On th thortial sid, ods on graphs provid svral intriguing prolms in th intrstion of oding thory and graph thory. In this papr, w study ods dfind y planar Tannr graphs. W driv an uppr ound on minimum distan d of suh ods as a funtion of th od rat R for R 5/8. Th ound is givn y ı 7 8R d + 7. (R ) Among th intrsting onlusions of this rsult ar th following: () planar graphs do not support asymptotially good ods, and () finit-lngth, high-rat ods on graphs with high minimum distan will nssarily non-planar. I. INTRODUCTION Th sptaular suss of ods on graphs has rsultd in immns rnt rsarh ativity on th pratial and thortial aspts of graphial ods. On th pratial sid, th powrful notion of rprsnting parity onstraints on Tannr graphs [][] has rsultd in trmndous simplifiations in th onstrution and implmntation of apaity-approahing ods for various hannls. On th thortial sid, th intrplay of graph thory and oding thory has rsultd in many intriguing prolms. In this papr, w ar onrnd with ods that ar dfind y planar Tannr graphs. Spifially, w study th minimum distan of ods that hav a planar Tannr graph. Planarity of a graph, a lassi notion in graph thory, allows for th mdding or rndring of a graph as a pitur on a twodimnsional plan with no two dgs intrsting. Spifi xampls of suh graphs ar trs and graphs with nonovrlapping yls. Intrstingly, oth ths typs of graphs hav n shown to orrspond to ods with poor minimum distan proprtis [][]. In this papr, w show similar proprtis for high-rat ods that hav planar Tannr graphs. Spifially, th main rsult of this papr is that a od of rat R 5/8 with a planar Tannr graph has minimum distan oundd as d 7 8R (R ) + 7, whr x (for a ral numr x) is th smallst intgr gratr than or qual to x. Not that th rsult holds for any loklngth. Hn, non-planarity is vital for larg minimum distan at high rats. This rsult provids justifiation for many known rsults on ods on highly non-planar graphs with larg minimum distan [5], and suggsts mthods for othr possil onstrutions. Th mthod of proof is novl and involvs svral stps. A givn planar Tannr graph is modifid through a sris of onstrution stps to a planar Tannr graph with maximum it nod dgr. For th modifid graph, th xistn of lowwight odwords is shown y an avraging argumnt. Th xistn is thn xtndd to th original Tannr graph. Th rst of th papr is organizd as follows. Th onstrution of th modifid Tannr graph is prsntd in Stion II. A simpl vrsion of th main rsult is provd in Stion III for th sak of larity in xposition. A omplt proof of th ound on minimum distan for ods on planar graphs is givn in Stion IV. Finally, onluding rmarks ar mad in Stion V. II. CONSTRUCTIONS Considr a Tannr graph G dfining a linar od. Th vrtx st of G is dnotd V (G) = V V, whr V and V dnot th st of it and hk nods of th G. W will assum that th rat of th od dfind y G is R = V / V. Lt Vi dnots th st of dgr-i it nods. Th dg st of G is dnotd E(G) V V, and an dg of G onnting it nod v to hk nod v is dnotd (v, v ). For a st of it nods V V, N (V ) dnots th st of hk nods onntd to V i.. N (V ) = v V {v : (v, v ) E(G)}. Th dgr of a it nod v is N (v ). For V V, th st of indud it nods I(V ) dnots th it nods whos nighors ar susts of V i.. I(V ) = {v : N (v ) V }. Th following proposition (statd without proof) onnts susts of hk nods and thir indud it nods to th minimum distan of th od. Proposition : Considr V V in a Tannr graph G dfining a od with minimum distan d. If I(V ) > V, thn d V +. Following Proposition, a sust of hk nods V is said to odword-supporting whnvr I(V ) > V. In this papr, w provid ounds for th minimum distan of Tannr graphs that ar planar i.. Tannr graphs that an mddd in a plan with no two dgs intrsting [6]. All Tannr graphs in th rst of th papr will planar with a fixd mdding. For planar Tannr graphs, w us Proposition for ounding minimum distan y showing th xistn of suital odword-supporting susts of hk nods. For this purpos, w dfin a nw planar graph involving th hk nods of th givn planar Tannr graph.

2 A. Chk graph of a planar Tannr graph Givn a planar Tannr graph G, th hk graph of G, dnotd C(G), is a planar graph with vrtx st V (th st of hk nods of G). W us an mdding of G in a plan, and pla th nods of C(G) in an isomorphi plan at th sam loations as th hk nods of G in th original plan. To aid in th onstrution, w idntify th loations of th it nods of G in th plan of C(G). Th dgs of C(G) ar onstrutd as follows: ) Considr a it nod v V with dgr λ > and N (v ) = {v,0, v,,, v,λ } lalld in a lokwis squn in th planar mdding i.. no dg out of v lis in v,i v v,(i+)λ ((x) λ dnots x mod λ). Add dgs (v,i, v,(i+)λ ) for 0 i λ to form a simpl yl nlosing th loation of v in th plan of C(G). ) In Stp, dgs ausing a fa nlosd y two dgs should not addd. ) Add mor dgs to mak th graph maximal planar (a planar graph is maximal if on mor dg will mak th graph non-planar [6]). Th onstrution of th hk graph is illustratd in Fig.. In Fig., th nods of th hk graph C(G) ar th hk a d G Fig.. Constrution of hk graph I. f f f C(G) nods of G dnotd {,,, }. In Stp, for it nod a of G, w onnt th nods {,,, } of C(G) in a yl. Stp, for th nods, and d of G with dgr largr than, rsults in fas with two dgs. Hn, no othr dgs ar addd to C(G) as pr Stp. In Stp, th dgs (,) and (,) ar addd to mak th hk graph maximal planar. Not that thr ar four fas in C(G), lalld f,, f and f in Fig.. Th fa f is th xtrior or xtrnal fa. In gnral, maximal planarization in Stp is not uniqu. Hn, thr an many hk graphs orrsponding to a singl Tannr graph. W fix on suh hk graph and all it th hk graph of G. Th following orrspondns twn a planar Tannr graph G and its hk graph C(G) ar vital for th minimum distan ounds. In Stp, a dgr- it nod v of G maps to a triangular fa in C(G) onnting th thr hk nods in N (v ). Hn, w say that a dgr- it nod is idntifid with a fa in C(G). In som ass, this an th xtrnal fa. In Stp, a dgr- it nod v of G maps to an dg in C(G) onnting th two hk nods in N (v ). W say that a dgr- it nod is idntifid with an dg in C(G). A dgr- it nod v of G dos not rsult in any dgs, ut v is rprsntd y th on hk nod N (v ) in C(G). W say that a dgr- it nod v is idntifid with th hk nod N (v ). In Stp, a it nod of dgr λ > rsults in a fa nlosd with λ dgs. In Stp, maximum planarization onvrts suh a fa into λ triangular fas. In th xampl of Fig., th dgr- it nod of G rsults in two triangular fas f and in C(G). Th hk nod orrsponds to th dgr- it nod, whil th dgs {(, ), (, ), (, )} orrspond to th dgr- it nods. Two mor xampls to illustrat th onstrution of th hk graph ar shown in Fig.. For th graph G in Fig. Fig.. a G a H Constrution of hk graph II. f C(G) f f f C(H), w gt a triangl around th loation of it nod a in Stp. For it nod, th dottd lins show th possil dgs in Stp. Howvr, no nw dgs ar addd as thy rsult in fas nlosd y two dgs. Noti that th irular dottd lin would hav rsultd in th xtrnal fa ing nlosd y two dgs. In G, th intrnal triangular fa f orrsponds to th dgr- it nod a, whil th xtrnal triangular fa orrsponds to. For th graph H in Fig., two triangls ar addd around th it nods a and in Stp. Not that th dg from hk nod to hk nod for it nod nds to drawn in a irular fashion to nlos th loation orrsponding to. Sin C(G) is maximal planar, y standard rsults in graph thory [6], w know that thr ar V fas in C(G) and all fas ar triangular (nlosd y thr dgs). Also, sin fas in C(G) rsult from it nods of dgr at last, w s that th maximum numr of it nods of dgr in a planar Tannr graph is limitd to V.

3 Th fas of C(G) ar dnotd F (C(G)). A fa f F (C(G)) is nlosd y thr dgs onnting thr hk nods of G. Th thr hk nods of G that form f ar dnotd V (f). In th xampl of Fig., w hav V (f ) = {,, }, V ( ) = {,, }, V (f ) = {,, }, and V (f ) = {,, }. B. Prlud To illustrat th usfulnss of th hk graph, w now prsnt a ound on minimum distan of rat 7/8 ods with a planar Tannr graph whos maximum it nod dgr is. Proposition : Lt G a planar Tannr graph with n it nods and m hk nods dfining a od with rat R = m/n 7/8 and minimum distan d. Lt th maximum dgr of a it nod in G. Thn, d. Proof: Lt f i, i m, th fas of th hk graph C(G). Lt w i = I(V (f i )) th numr of it nods indud y th st of hk nods V (f i ) forming th fa f i. Lt f k th fa suh that w k w i, i m. Sin ah it nod of G has dgr at most, it is indud at last on y som fa f i in C(G). So, w hav w + w + + w m n. Sin w k w i and n 8m, w simplify as follows. w k (m ) n, n w k m > n m. Hn, w k = I(V (f k )) 5 and V (f k ) =. Thus, thr xists a ( 5, ) suod of th original od. This implis that d. Th aov proof uss th fas of C(G) to onstrut odword-supporting st of hk nods in th original Tannr graph G. An avraging argumnt is usd to show th xistn of th odword-supporting st. Ths thms will usd and xtndd in th rmaindr of this papr to prov mor ounds on th minimum distan of ods with a planar Tannr graph. Anothr ruial assumption in Proposition is on th maximum it nod dgr. This assumption will rlaxd through anothr onstrution alld th hk invrs. C. Chk invrs of a planar Tannr graph Th sam hk graph an rsult from svral planar Tannr graphs. Fig. illustrats on suh xampl whr two Tannr graphs G and H rsult in th sam hk graph C(G). Fig.. G C(G) = C(H) On hk graph for multipl Tannr graphs. H Givn a hk graph C(G) of a planar Tannr graph G, w an onstrut a spial planar Tannr graph G with maximum it nod dgr suh that C(G ) = C(G). Th onstrution of this spial Tannr graph, whih w all th hk invrs of G, is dsrid nxt. Givn a planar Tannr graph G, th hk invrs of G, dnotd G, is a planar Tannr graph with hk nod st V = V and it nod st V = V V V onstrutd as follows: ) V = F (C(G)). A it nod v V orrsponds to a fa f F (C(G)), and is onntd to th thr hk nods in V (f) that form th fa f in C(G). Bit nods in V hav dgr, and V = V. ) Th st V V is an aritrary sust of V V (th st of dgr- and dgr- it nods of G) of siz V V = [ V ( V )] + whr [x] + = max(x, 0). A it nod v V V V V is onntd to th hk nod(s) in N (v ). Exampl : Th onstrution of th hk invrs for a planar Tannr graph G is illustratd in Fig.. Th Tannr Fig.. f h a d G f f g i f d G Constrution of hk invrs. f f f i C(G) = C(G ) f graph G has 9 it nods ( of dgr, of dgr, of dgr ) and hk nods. Th hk graph C(G) has nods, orrsponding to th hk nods of G, and fas ( intrior and xtrior) nlosd y thr dgs ah. Th hk invrs G has hk nods orrsponding to th hk nods of G or th nods of C(G). Th it nods of dgr in G orrspond to th fas of C(G) from Stp of th onstrution of G. Ths nods ar lalld with th lals of th fas in C(G). In Stp, 5 it nods of dgr- and (th nods,, d, f, i ) ar addd to G with onntions aording to th orrsponding onntions in G. In Fig., th dottd lins in G ar th dgs of C(G) = C(G ).

4 Th following proprtis of th hk invrs of a planar Tannr graph ar important for futur onstrutions: If V > V, th hk invrs G has th sam numr of hk nods and it nods as G. If th rat of th od dfind y G is gratr than /, w hav V > V. For rat gratr than /, w hav V + V = V ( V ). () From now on, w will rstrit ourslvs to planar Tannr graphs G that dfin ods of rat gratr than / so that () always holds. Th hk graph of G is sam as C(G) i., C(G ) = C(G). But ah fa in C(G ) orrsponds to a dgr it nod in G unlik in C(G) and G. Th hk invrs plays a ruial rol in th minimum distan ounds. Distan ounds will first shown for th od rprsntd y th Tannr graph G. Thn, th sam ound will sn to hold for th original graph G. D. Dual of hk graph and odword-supporting sugraphs Sin th hk graph C(G) is planar, w an dfin its dual as dfind for any planar graph [6]. Sin w ar working on a partiular mdding, th dual graph is uniqu. Th dual of C(G), dnotd as C (G), is a planar graph with vrtx st V (G) that has a on-to-on orrspondn with F (C(G)). Two vrtis of C (G) ar joind y an dg whnvr th orrsponding fas of C(G) shar an dg. Sin C(G) is maximal planar with V vrtis and V triangular fas, C (G) has V vrtis ah of dgr. Sin thr is a on-to-on orrspondn twn dgs in a planar graph and its dual, lt us dnot th dg orrsponding to in C(G) as in C (G). If thr ar multipl dgs twn any two nods of C (G), w an show that th minimum distan of th od rprsntd y G is at most (S Stion IV-F for a proof). So, w onsidr planar Tannr graphs G and hk graphs C(G) suh that thr ar no multipl dgs in C (G). Sugraphs of th dual of th hk graph play an important rol in dtrmining th xistn of low-wight odwords in th od (or small odword-supporting susts of hk nods in th od s Tannr graph). Th asi ida is th following. Using a vrtx-indud sugraph of th dual of th hk graph C (G), w dfin susts of hk nods of G and study whn thy ar odword-supporting. Lt U V (G) = F (C(G)) indu a sugraph C U (G) of C (G). Baus of th ongrun, w will dnot a vrtx of C (G) as a fa f F (C(G)). Hn, U F (C(G)). For ah sust U, w assoiat a sust of hk nods VU of G as givn low: VU = f U V (f), () whr V (f) (as for) is th st of thr nods that form th fa f in C(G). Hn, vry vrtx sust U in C (G) orrsponds to a sust of hk nods VU in G. W will prov xistn of odword-supporting susts among th sts VU produd y diffrnt U. Th sugraph of C(G) indud y VU is dnotd C U (G). In spit of th notation, not that C U (G) is not nssarily th planar dual of C U (G). As for, th st of it nods indud y VU in th Tannr graph G is dnotd I(VU ). Th wight of an indud sugraph C U (G), dnotd wt(c U (G)) or wt(u), is dfind as wt(u) = I(VU ), th numr of indud it nods of V U. An indud sugraph C U (G) of C (G) is said to odword-supporting if wt(u) > VU. Exampl : Considr a planar Tannr graph G with V = {a,,,d,,f,g} and V = {,,,, 5} as shown in Fig. 5. Th hk graph C(G) has 5 vrtis orrsponding to 5 Fig. 5. d f G a 00 g f f f f 00 f 6 Dual of th hk graph C (G) f 6 f f f f 5 C(G) hk nods of G and 6 fas (inluding th xtrnal fa). Th 6 vrtis in dual C (G) ar lalld aording to th orrsponding fas in C(G). Lt U = {f, f } V (G). Thn, VU = {,,, } and C U (G) = f f f. Th it nods indud y VU, I(V U ) = {a,,,d} and hn wt(u) =. In Fig. 5, th dgr- it nods of G {,d,} ar idntifid with th rsptiv fas {f, f, f 5 } in th hk graph C(G) and th orrsponding vrtis in th dual of th hk graph C (G). Th dgr- it nods of G {,f,g} ar idntifid with th rsptiv dgs {(, ), (, 5), (, 5)} in C(G) and th orrsponding dgs {(f, f 6 ), (, f 5 ), (, f 6 )} in C (G). Th dgr- it nod {a} of G is idntifid with nod of C(G) and th xtrnal fa of C (G). In th main rsult of this papr, w stalish th xistn of odword-supporting sugraphs indud y a small sust U in th dual of th hk graph of a planar Tannr graph. Th nxt two propositions show that th siz of VU and th minimum distan of th od ar oundd y th siz of U that indus a odword-supporting sugraph in C (G). Hn, small odword-supporting sugraphs in th dual of th hk graph rsult in low-wight odwords in th od. Proposition : Lt C U (G) a onntd indud sugraph of C (G) indud y a propr sust U F (C(G)). Thn, for 5

5 5 th sust of vrtis U, V U U + (U), whr (U) is th numr of simpl yls in C U (G). Proof: As shown in Appndix A, th sugraph C U (G) an onstrutd y adding nods on at a tim from th st U in a suital ordr. Th ordr is suh that, at ah stp of th onstrution, th most rntly addd nod has dgr or aftr inlusion. Hn, th rsulting sugraph aftr ah stp is onntd. With this partiular ordring of nods of U, w will prov th proposition y indution. Lt U =. Thn, VU = = U + 0. Hn, th proposition is tru for U =. Assum that it is tru for th sugraph indud y P nods whr P U i.. V P P + (P ), whr (P ) is th numr of simpl yls in C P (G). W will prov that th rsult holds whn a nw nod w is addd. Considr th sugraph C W (G) indud y W = P w. By th ordring of th nods, w has dgr or in C W (G). Cas : w has dgr. Sin w is onntd to C P (G) y xatly on dg, th fa in C W (G) orrsponding to w rsults in th addition at most on nw nod to C P (G). Also, th addition of w dos not rat a nw yl in C W (G). Hn, V W V P + P + (P ) + = W + (W ), whr (W ) is th numr of simpl yls in C W (G) whih is qual to (P ). Cas : w has dgr Sin w is onntd to C P (G) y two dgs, th nods at th oundary of th fa orrsponding to w in C W (G) ar alrady prsnt in C P (G). Thus numr of nods in C P (G) is sam as that in C W (G). Also, th addition of w inrass th numr of yls in th rsulting sugraph y. This is aus w onnts two nods in P that ar alrady onntd in C P (G). Hn, V W = V P P + (V ) + = W + (W ) whr (W ) is th numr of simpl yls in C W (G). Th two ass ar illustratd in Fig. 6. Th dottd lins in Fig. 6 ar th dgs indud in C (G) y th addition of th nod w. Thus y indution, V U U + (U). Proposition : Lt G a planar Tannr graph of a od with minimum distan d. If thr is a odword-supporting onntd sugraph of C (G) indud y a propr sust U, thn d U +. Proof: Sin C U (G) is odword-supporting, wt(u) = I(VU ) > V U. By Proposition, d V U +. Sin C U (G) is onntd, y Proposition, VU = U + U +. Thrfor, d U +. Thrfor, to ound minimum distan, w sarh for sugraphs of C (G) on minimum numr of vrtis that ar odword-supporting. Fig C P (G) w 00 nw nod Cas C P (G) 00 Cas w C P (G) no nw nod On stp in indutiv onstrution of C U (G). III. DISTANCE-RATE BOUNDS w no nw nod Th main rsult of this papr is th following thorm. Thorm 5: Lt G a planar Tannr graph rprsnting a od with rat R 5/8 and minimum distan d. Thn, d 7 8R (R ) +. Th proof involvs multipl stps. In th first stp, w onstrut th hk invrs G, hk graph C(G ) = C(G) and its dual C (G ) as disussd in Stion II. In th sond stp, th distan ound of Thorm 5 is shown for th od orrsponding to th hk invrs G y proving th xistn of a suital odword-supporting sugraph in C (G ). In th third and final stp, th sam ound is shown to hold for G y a sris of graph manipulations. For larity of xplanation, w first show th sond and third stps in th proof for a wakr vrsion of Thorm 5. For th wakr vrsion, th odword-supporting sugraph of C (G ) is simply an dg. Howvr, th important idas in th gnral proof ar prsnt in th wakr vrsion as wll. A gnral proof of Thorm 5 is prsntd latr in Stion IV. A. Illustrativ proof A wakr vrsion of Thorm 5 is th following. Thorm 6: Lt G a planar Tannr graph rprsnting a od with rat R /6 and minimum distan d. Thn, d 5. W first prov a fw lmmas that ar usd in th final proof. Using th onstrutions in Stion II, lt G th hk invrs of G. Lt C(G ) th hk graph of G and lt its dual C (G ). ) Codword-supporting sugraph for G : Lmma 7: Lt G a hk invrs of G supporting a od of rat R /6. Thn, thr is an dg in C (G ) that is odword-supporting.

6 6 Proof: Considr an dg = (f, ) E (G ), whr E (G ) dnots th dg st of C (G ). Sin C(G ) is maximal planar on V vrtis, w hav E (G ) = V 6. W st U = {f, } and gt C U (G ) = as shown in Fig 7. For simpliity, th st U is rplad with in th notation. For instan, VU will dnotd V and so on. f Fig. 8. Computing q(v ) for a dgr it nod v. f f Fig. 7. C U (G) Edg as th indud sugraph. C U (G) Considr th following summation: Y(G ) = ( wt( ) V ). () E (G ) W will show that Y(G ) > 0, whih implis that thr xists an dg suh that wt( ) > V proving th lmma. From Fig. 7, V = for all. So, to valuat Y(G ), w writ E (G ) wt( ) as follows: wt( ) = q(v ), E (G ) v V whr q(v ) is th numr of tims a nod v of G is indud y V, E (G ). Lt η i = q(v ) () whr V i v V i is th st of dgr-i it nods in G. Thn, wt( ) = η + η + η E (G ) W an now valuat q(v ) in th trms η i for i =,,. A dgr- it nod v V orrsponds to a triangular fa in C(G ), whih orrsponds to a dgr- nod f C (G ). Whnvr an dg is inidnt on f, th nod v will indud y V. Sin thr ar dgs inidnt on any nod in C (G ), q(v ) = for v V. A dgr-two it nod v V is idntifid with an dg in C(G ). Not that this dg is ommon to two fas, say f and in C(G ). Lt f and th orrsponding nods in C (G ). Thn, v is indud y V, whnvr is inidnt to f or. Sin thr ar 5 dgs inidnt on two nighoring nods f and in C (G ), q(v ) = 5 for v V. This is illustratd in Fig 8. A dgr- it nod v V in G is idntifid y a hk nod to whih it is onntd to in G. This hk nod orrsponds to a fa in C (G ). Th nod v is indud y V, whnvr is inidnt on on or mor vrtis of th fa. Sin thr ar at last vrtis in a fa, not that q(v ) for a dgr- it nod v is gratr than q(v ) for a dgr- it nod v. Thus, wt( ) V + 5 V + 5 V, E (G ) upon using (). Using in (), = ( V ) + 5( V ( V )), = 5 V V + 8, Y(G ) 5 V V + 8 ( V 6), = 5 V 6 V +. W s that Y(G ) > 0, whnvr R = V V 6. By Lmma 7, w hav shown that thr is a odwordsupporting dg in C (G ). ) Codword-supporting sugraph for G: To xtnd th proof to a gnral planar Tannr graph, w show that a sris of simpl modifiations an transform th hk invrs G to th original Tannr graph G. W gin y dfining thr asi oprations on a gnri planar Tannr graph P. ) DS: Rmov a dgr- it nod in P and add a dgr- it nod to som hk nod. ) DS: Rmov a dgr- it nod in P and add a dgr- it nod to a pair of hk nods kping th rsulting graph planar. ) DE: Inras th dgr of a dgr- it nod y onnting it to on or mor hk nods so that th rsulting graph is still planar. Th rsulting inras in dgr is alld th xpansion fator of DE. Th arviation DS stands for Dgr Shrinking, and DE stands for Dgr Expansion. Ths oprations ar illustratd in Fig 9. In Fig. 9, th solid lins ar th dgs of th Tannr Fig. 9. a a a DS DE Rmov Add Illustration of DS and DE oprations. Inras dgr of a graph and th dottd lins ar dgs of th hk graph. Not that th hk graph is unaltrd y th DS and DE oprations. Proposition 8: Lt G a planar Tannr graph with hk invrs G. Thn, G an otaind from G y a sris of DS, DS and DE oprations.

7 7 Skth of proof: In th pross of onstruting G, th following osrvations an mad: () dgr- nods of G ar rtaind in G, () som dgr- and dgr- nods may droppd, and () highr dgr ( ) nods of G rsult in multipl dgr- nods in G. Th oprations DS and DS rstor th droppd dgr- and dgr- nods, whil a following DE opration rats highr dgr nods. Not that DS and DS rat mpty fas in th hk graph of G whil DE maks a st of fas orrspond to a singl it nod of highr dgr. Also not that if w start with G, DE y a fator of x is always prdd y x DS oprations sin x mpty fas should ratd for DE in ordr to prsrv planarity. Ths oprations rursivly position dgr and dgr it nods to th positions as in G and also rat it nods of highr dgr mathing to thos in G. Exampl : Considr th Tannr graph G and its hk invrs G in Fig. of Exampl. Starting with G w an otain G through a sris of DS, DS and DE oprations. This is illustratd in Fig. 0. Not that C(G) rmains a valid hk graph of th rsulting graph aftr vry opration. G f f f C(G) = C(G ) DS DS DS W ar now rady to prov th xistn of a odwordsupporting dg in C (G). Th approah is to show that th oprations DS, DS and DE annot dras Y(G ). Hn, at th nd of th nssary sris of DS and DE oprations to gt G from G, w hav Y(G) > 0. Th rsult is provd in th following lmma. Lmma 9: Lt G a planar Tannr graph rprsnting a od with rat R /6. Thn thr is an dg in C (G) that is odword-supporting. Proof: W will prov y showing that th oprations DS and DE on th it nods of G to gt G ar suh that Y(G) Y(G ) (s () for dfinition). Sin oth C (G ) and C (G) hav sam strutur whn sn as graphs, th trm E (G ) V = E (G) V. Th hang will in E (G ) wt( ). Lt = wt( ) wt( ). E (G) E (G ) W will show that 0 to laim th lmma. Lt H th planar graph otaind at som intrmdiat stp in th transformation from G to G. Lt us s how DS and DE oprations afft E (H) wt( ). Th opration DS rdus th numr of dgr- it nods y on, and inrass th numr of dgr- it nods y on. Lt η and η th nw valus of th trms η and η in () aftr th opration DS. Lt δ DS th hang in E (H) wt( ) whn a DS is prformd. W s that η = η η = η + 5 δ DS = (η + η ) (η + η ) = > 0 Sin DE y a fator of x is prdd y x DS oprations, w will study th nt fft. DE with xpansion fator of x prdd y x DSs rsults in th following: (i) rdus th numr of dgr- it nods y x + (ii) inrass th numr of dgr- it nods y x (iii) introdus a it nod of dgr x +. Efft of (i) and (ii) an radily drivd as for. As (iii) involvs introdution of a it nod, it inrass Y(H) y a positiv quantity, say α x. Lt δ DE th hang in E (H) wt( ) whn a DE is prformd. Thrfor, δ DE = (x + ) + 5(x) + α x. (5) DE G Fig. 0. Illustration of Proposition 8. Sin δ DE is non-ngativ for x >, it is nough to omput α and show that δ DE is non-ngativ for x =. Whn th xpansion fator is on, a dgr- it nod v oms a dgr- it nod, and v is idntifid with two fas of th nw hk graph having a ommon dg. This is quivalnt to saying that v is idntifid with an dg in th dual of th hk graph. Thrfor, α =. Sustituting in (5), w gt δ DE = 0 for x =. As q(v ) of a dgr- it nod v is mor than q(v ) of a dgr- it nod v, δ DS is non-ngativ whnvr DS is usd in pla of DS whr δ DS is th hang in E (H) wt( ) whn a DS is prformd.

8 8 Sin G an otaind from a sris of DS, DS and DE oprations, and sin δ DS, δ DS, and δ DE ar all nonngativ, 0. Hn, th lmma is provd. ) Proof of Thorm 6: By Lmma 9, C (G) has an dg suh that I(V ) > V. W s that V = y Proposition. Hn, y Proposition, th od dfind y G has a odword of wight at most V + = 5. This provs Thorm 6. IV. PROOF OF MAIN RESULT W now provid th proof of Thorm 5. Th mthod and stps of proof ar similar to that of th proof of th wakr Thorm 6. A odword-supporting sugraph will shown to xist in th dual of hk graph of hk invrs y a similar ounting argumnt. Th rsult will thn xtndd y DS and DE oprations to th original graph. Th main hang is that an dg of th dual nd not odword-supporting for lowr rats. W will show that among th nighorhoods of vrtis of th dual of hk graph with p nods (for a suitaly hosn p), thr xists a odwordsupporting sugraph, whih provids a ound on minimum distan. W will impos a girth ondition on th dual of hk graph, so that th nighorhoods om trs for small p. Sin th dual of hk graph is rgular with dgr, th vrtx nighorhoods will rootd trs with mostly dgr- nods xpt nar th lavs. Th girth ondition on th dual of th hk graph will latr shown to translat into a ondition on th rat of th od dfind y th original Tannr graph. Th typ of nighorhood strutur in th dual hk graph is apturd y -trs dfind low. A. -trs A -tr rootd at a vrtx v r is a rootd tr in whih th root v r has at most hildrn and all othr nods hav at most hildrn. Th dpth of a vrtx v is th lngth of th path from th root v r to th vrtx v. Th st of all nods at a givn dpth is alld a lvl of th tr. Th root nod is at dpth zro. Th hight of a tr is th lngth of th path from th root to th dpst nod in th tr. A omplt -tr is on in whih vry lvl, xpt possily th last, is ompltly filld. Th numr of nods at lvl l (xpt possily th last) of a omplt -tr is. l and numr of nods up to and inluding lvl l is. l. Fig. shows a pitur of a omplt -tr. l = 0 l = l = l = v r Fig.. A Complt -tr rootd at v r. Hr ar a fw rsults on omplt -trs. h = ) Lt p th numr of vrtis of a omplt -tr. Thn p =. l(p) + z(p) for a suital lvl l(p) suh that 0 z(p) <. l(p), whr z(p) dnots th numr of nods in th last lvl. ) Hight of th omplt -tr with p vrtis is { l(p), z(p) 0 h(p) =. l(p), z(p) = 0 ) Thr ar = ( ). l(p) z(p) omplt -trs of hight h(p) rootd at a givn vrtx v. Eah suh tr is alld a ralization of th omplt -tr rootd at v. ) Thr ar thr ranhs from th root of a omplt - tr on p nods. Th numr of nods up to lvl l < h(p) on on ranh (xluding th root nod) is (.l ) = l. Th paramtrs l(p), z(p), h(p) and ar valuatd for som valus of p in Tal I. p l(p) z(p) h(p) TABLE I PARAMETERS OF A COMPLETE -TREE FOR SOME VALUES OF p B. Complt -graphs in C (G) Sin C (G) is planar with uniform dgr, w an look for omplt -trs on p vrtis in th vrtx nighourhoods of C (G). Givn p, ah root has omplt -trs of hight h(p) only whn th girth g of C (G) satisfis g h(p) +, (6) whr h(p) is th hight dfind as for. Lt V (G) th vrtx st of C (G). Lt V i,j V (G) th vrtx st of th j-th ralization of a omplt -sutr rootd at nod v i with p vrtis. Lt th sugraph indud y V i,j dnotd as C i,j (G) for simpliity. Ths indud sugraphs from now on will rfrrd to as omplt - graphs. Not that C i,j (G) nd not a tr du to th prsn of on or mor xtra dgs that an rat yls. Howvr, undr th girth ondition, ths additional dgs an only twn lavs of th omplt -tr rating yls of lngth h(p) +. Ths additional dgs twn th lavs of a ralization of a omplt -tr in th nighorhood of a nod ar said to yl-rating with rspt to that partiular ralization. Th numr of ralizations with root v i in whih an dg is yl-rating is alld th rurrn numr of th dg with rspt to th vrtx v i, and is dnotd as r vi ( ). Th total numr of all ralizations with rspt to all roots

9 9 in whih is yl-rating is alld th total rurrn numr of th dg and dnotd r( ). W radily s that r( ) = r vi ( ). (7) v i V (G) An dg that is yl-rating for at last on ralization (or dg with positiv total rurrn numr) is alld a yldg. Th st of all yl-dgs in C (G) is dnotd y B(G). Exampl : Considr th dual graph C (G) shown in Fig.. W will show som omplt -graphs in C (G) of Fig. a f d 5 G g f 6 C (G) f f f f f f f 5 f f 5 f 6 (a) () () f f 8 7 f 5 9 f 6 5 Fig.. An illustration of omplt -graphs rootd at fw nods for p =. for p =. Fig. (a) shows a omplt -graph rootd at f in whih is a yl-rating dg. Similarly, (),() shows omplt -graphs rootd at f and rsptivly. Th orrsponding yl rating dgs ar shown in dashd lins. Osrv that whih is yl-rating for th omplt - graph rootd at f, ut is not yl-rating for th omplt -graph rootd at f. For th omplt -graph rootd at f, is yl-rating. Sin p =, = i., thr is only on ralization pr root nod. Hn, r f ( ) =. Also not that is not ylrating for othr omplt -graphs rootd at othr nods. Thrfor, r( ) =. Using similar argumnts, w an show that r( i ) = for i = {,, 5, 6, 7}. Ths dgs ar yldgs in C (G). All othr dgs ar non-yl-dgs; hn, thir rurrn numr is zro. Lt W vi,j = wt(c i,j (G)) th wight of th j-th ralization of a omplt -graph rootd at v i. Lt Vi,j th st of hk nods forming th fas orrsponding to th vrtis V i,j in C (G) i.. Vi,j = V U with U = V i,j as in (). By Proposition, Vi,j p + vi,j, (8) whr vi,j is th numr of yls in th j-th ralization of th omplt -graph on p nods rootd at v i. By dfinition, C i,j (G) is odword-supporting if W v i,j Vi,j > 0. Considr th summation, Y(G) = v i V (G) j= ( Wvi,j V i,j ) (9) whr is th numr of ralizations of omplt -trs on p vrtis rootd at a partiular nod assuming th girth ondition (6). W will show that Y(G) > 0 to stalish th xistn of a odword-supporting sugraph among th omplt -graphs of C (G). Lt m = V and n = V th numr of hk nods and it nods of G. Hn, V (G) = m. Using (8) in (9), w gt Y(G) v i V (G) j= W vi,j [(m )p + (m )] + v i V (G) j= vi,j, (0) To show Y(G) > 0, w simplify th trms in th right hand sid of (0). to rur- C. Simplifying v i V (G) j= v i,j W gin y rlating th numr of yls vi,j rn numr of dgs. Proposition 0: Th following quality holds: v i V (G) j= Proof: W s that v i V (G) j= vi,j = vi,j = v i V (G) l B(G) r( l ) l B(vi) r vi ( l ), whr B(v i ) is th st of dgs that ar yl-rating in any of th omplt -graphs rootd at v i. Now, v i V (G) j= vi,j = l B(G) v i V (G) r vi ( l ) = r( l ). () l B(G) ) Oupid and unoupid dgs: In th omputation of Y(G), th yl dgs idntifid with dgr- it nods should tratd sparatly. Suh yl dgs ar lassifid nxt. Eah dgr- it nod in G is idntifid as an dg in C(G). An dg in C(G) is said to oupid y a dgr- it nod if th it nod is onntd to th hk nods of. Els it is said to unoupid. Sin thr is a on to on orrspondn twn dgs of C(G) and dgs of its dual C (G), w us th trms oupid and unoupid for dgs of dual as wll. Similarly, w an talk of oupid and unoupid hk nods. A hk nod of a planar Tannr graph is said to oupid if thr is a dgr- it nod onntd to it. Th dgr- it nod is said to oupy th hk nod. Othrwis, th hk nod is said to unoupid. Lt B(G) th st of yl-dgs, and lt B o (G) and B u (G) th st of oupid and unoupid yl-dgs.

10 0 W s that B o (G) and B u (G) partition B(G) so that th following holds: B(G) = B o (G) + B u (G). () Exampl 5: In th Tannr graph of Figs. 5 and, th st of oupid dgs ar givn y {, 6, 7 } as sn from Exampl, and th st of yl-dgs is givn y B(G) = {,, 5, 6, 7 }. Hn, w s that B o(g) = { 6, 7 } and B u (G) = {,, 5 }. Using th partition of B(G) in (), w s that v i V (G) j= vi,j = r( l ) + r( l ). () l Bo(G) l Bu(G) Exampl 6: In th illustration of th DS and DE oprations in Fig. 0, w statd that th hk graph of G and G ar th sam. Howvr, th st of oupid dgs and hk nods hangs aus of th hangs in th numr of dgr- and dgr- nods. In th hk graph C(G ) in Fig. 0, th st of oupid dgs is {(, ), (, ), (, )}, and th st of oupid nods is {, }. Howvr, in C(G) in Fig. 0, th st of oupid dgs is {(, ), (, ), (, ), (, )}, and th st of oupid nods is {,,, }. Sin nw dgr- and dgr- nods an possily addd in G through th DS oprations, som unoupid dgs and hk nods in C(G ) om oupid in C(G). ) Singular nods: Anothr spial situation ariss with multipl dgs in hk graphs. Not that, y onstrution, multipl dgs an aris in hk graphs, if thy do not rsult in a fa nlosd y two dgs. Th fft of multipl dgs in hk graphs is haratrizd nxt. Dfinition: Lt G a planar Tannr graph, and lt C(G) its hk graph. Lt C (G) a sugraph of C(G) onsisting of a maximal planar graph on vrtis plus an dg that lads to an xtrnal fa of lngth in C (G) as shown in Fig.. In addition, w impos th onstraint that th intrior fas of C (G) ar fas in C(G). A dgr- it nod in G is said to Fig.. Singular nods. 5 v C (G) singular if it oupis a hk nod in G that orrsponds to on of th intrior nods of C (G). In Fig., a dgr- it nod that oupis v or v is singular. Th numr of singular nods in a planar Tannr graph G is dnotd s G. Th following proposition rlats s G to th rurrn numr. v Proposition : Lt G a planar Tannr graph with s G singular nods. Lt B u (G) th st of unoupid yldgs in C (G) for th omplt -graphs on nods. If l Bu(G) r( l ) < s G, thr xists a odword-supporting sugraph on nods in C (G). Proof: Lt l Bu(G) r( l ) < s G. Sin l Bu r( l ) is non-ngativ s G is at last. Hn, thr is at last on sugraph of th form C (G) as shown in Fig. in C(G). Lt B th st of dgs in C (G) orrsponding to th st of intrior dgs {,,,, 5 } in C (G). Th dgs of B ar shown as dashd lins in Fig.. Not ths th dgs ar yl-dgs for p =. Fig.. Cyl dgs in th dual for p =. Lt C i (G), i =,...k th sugraphs in C(G) of th form C (G). Lt s i G th numr of singular nods in Ci (G) and B i th st of 5 yl-dgs orrsponding to th intrior dgs in C i (G). Osrv that B i ar disjoint. Lt B i u th sust of dgs of B i that ar unoupid in C (G). Lt B u = k i= Bi u. Thn, B u B u (G). Hn, l B u r( l ) k l Bu(G) r( l ) < s G, i= l Bi u k i= l Bi u r( l ) < k s i G, i= r( l ) si G < 0. Thus, thr xists j, j k, suh that, r( l ) < sj G, l Bj u B j u l Bj u r( l ) < sj G, B j u < s j G. Thus, thr ar at most s j G dgs in Bj that ar unoupid. In othr words, thr ar at last 6 s j G oupid dgs in Bj. Thrfor, th nods in C j (G) indu at last 6 s j G +sj G = 6 it nods in G. Sin, thr ar fas in C j (G), thr xists a odword supporting sugraph on nods in C (G).

11 D. Minimum distan ound for G Th nxt lmma is a gnralization of Lmma 7 from dgs to omplt -graphs for th main rsult. Lmma : Lt G a planar Tannr graph with hk invrs G. Lt G dfin a od with rat R. Thn, thr xists a odword-supporting sugraph on p vrtis in C (G ) whr 7 8R p = (R ) is suh that th girth ondition (6) is satisfid. As for, w will prov th rsult y showing that Y(G ) > 0. Simplifying v i V (G ) j= W v i,j: Lt us now onsidr th summation of wights trm in (0). As don in th illustrativ proof of Thorm 6, w lt v i V (G ) j= W vi,j = v V q(v ), whr q(v ) is th numr of tims a nod v of G is indud y Vi,j for all i and j. Lt η i = q(v ), () whr V i v V i is th st of dgr-i it nods in G. Thn, v i V (G ) j= W vi,j = η + η + η. (5) W will gin with alulation of η. A dgr- it nod v in G is idntifid with a nod f in C (G ). So, q(v ) is sam as th numr of omplt -graphs that ontain f, whih quals p. Hn, w s that Thrfor, q(v ) = p for v V η = (m )p. (6) Th omputation of η and η, as shown in Appndis B and C, rsults in th following. η = V ( p + ) r( l ) (7) l Bo(G ) η V ( p + ) s G (p), (8) whr s G (p) is givn y, { s G whn p = s G (p) = 0 ls (9) with s G ing th numr of singular nods in G as disussd in Stion IV-C. Using (6), (7) and (8), w gt v i V (G ) j= W vi,j (m )p + V ( p + ) + V ( p + ) r( l ) s G (p). l Bo(G ) Using V + V = n (m ), w gt v i V (G ) j= W vi,j [ pn+ n p(m ) (m )] l Bo(G ) r( l ) s G (p). (0) Proof of Lmma : Using () and (0) in th xprssion for Y(G ) in (0), w gt Y(G ) ( pn + n p(m ) 8 (m )) + r( l ) s G (p). () l Bu(G ) Lt X = ( pn + n p(m ) 8 (m )). Hn, Y(G ) X + r( l ) s G (p) () l Bu(G ) W fix p to th smallst intgr that rsults in X > 0. This is radily sn to 7 8R p =. () (R ) Cas : p. Whn p givn y () is not qual to, s G (p) = 0 y (9) and hn th trm l Bu(G ) r( l ) s G (p) is non-ngativ. This implis that, for this p, Y(G ) > 0 and thr xists a odword-supporting omplt -graph on p = (R ) nods. Cas : p =. If l Bu(G ) r( l ) s G (p) 0, Y(G ) > 0 for p = and thr xists a odword-supporting omplt -graph on (R ) p = = nods. Evn othrwis, y Proposition, thr xists a odword supporting sugraph on nods. E. Minimum distan ound for G Th ound for G is otaind as in th proof of Thorm 6 y showing that th oprations DS and DE do not rdu th valu of Y(G ). Th graph G is otaind from G through a sris of rursiv oprations as shown in Proposition 8. W will s how th DS and DE oprations involvd in transforming G to G afft th summation Y(G ). Sin oth C (G ) and C (G) hav sam strutur whn sn as graphs, th trm v i V (G) j= V i,j = v i V (G ) j= W v i,j. j= V i,j. Th only hang will in v i V (G ) Lt H a planar graph otaind at som intrmdiat stp in th transformation from G to G. Sin C(G ) is a hk graph for H, th girth ondition is satisfid y C (H). W writ W (H) = v i V (H) j= whr η i (H) = v Vi dgr-i it nods in H. W vi,j = η (H) + η (H) + η (H), q(v ) with V i () ing th st of

12 ) Efft of DS: Th graph otaind aftr th DS opration on H is dnotd DS[H]. Lt DS = W (DS[H]) W (H) th hang in th wight summations aus of th DS opration. Similar notation is usd for th DS and DE oprations. Th opration DS rdus th numr of dgr-thr it nods y on and inrass th numr of dgr- it nods y on. From (6) and (7), η (DS[H]) = η (H) p, ( η (DS[H]) = η (H) + p + ) r( ), whr is th dg in C (DS[H]) idntifid with th nw dgr- it nod. Hn, DS = δ r( ), (5) whr δ = p. + > 0. ) Efft of DS: Th opration DS rdus th numr of dgr- it nods y on and inrass th numr of dgr- it nods y on. From Appndix C, q(v ) for a dgr- it nod v is at last ( p + ) xpt for th as whn p = and v singular. Following alulations as for, w an show that DS = δ s (p), (6) { whr s, p = and v singular, (p) = 0, ls. ) Efft of DE: Lt DE(x) dnot th DE opration y an xpansion fator of x, whih is nssarily prdd y x DS oprations. Th opration DE(x) rsults in th following: (i) rdus th numr of dgr- it nods y x +. (ii) inrass th numr of it nods of dgr y x. (iii) introdus a it nod of dgr x +. Lt E th sust of dgs of C (H) idntifid with th nw dgr- it nods, and lt s + th numr of nw dgr- it nods that ar singular in DE(x)[H]. Efft of Stps (i) and (ii) an radily drivd as for. Th introdution of a it nod in Stp (iii) inrass W (H) y a positiv quantity dnotd α x. Hn, w s that ( DE(x) = x p + ) p. r( ) s + (p) + α x, whr s + (p) = B { s +, p =, 0, ls, and B is th st yl dgs in E. In th aov quation, w us B for th summation trm instad of E as rurrn numr of a non yl-dg is zro. Lt δ DE (x) = x ( p + ) p + α x. Thn, DE(x) = δ DE (x) r( ) s + (p). (7) E Sin α x is positiv, δ DE (x) is positiv for x. As shown in Appndix D, ( α p ). Sustituting α in δ DE (x) for x = givs δ DE () 0. As shown in Appndix D, ( α p ) 6 z(p). Sustituting α in δ DE (x) for x =, w gt ( δ DE () 6 z(p) + ) 6 p > 0, sin p > z(p). Hn δ DE (x) 0 for all x. ) Codword-supporting omplt -graph: W now prsnt a gnralization of Lmma 9 for omplt -graphs. Lmma : Lt G a planar Tannr graph dfining a od of rat R. Thn, thr xists a odword-supporting omplt (R ) -graph on p = nods in C (G) providd th girth ondition is satisfid. Proof: W will prov th lmma y showing that Y(G) > 0 for p = (R ). Lt n DS and n DS th numr of DS and DS oprations (xluding thos DS and DS oprations that ar prformd to rat mpty fas for DEs) prformd in otaining G from G. Lt n DE (x) th numr of DE oprations with xpansion fator x. Lt E th sust of dgs of C (G ) idntifid with th nw dgr- it nods rsulting from ths oprations (s Exampl 6). Lt s th numr of singular nods in G that ar not prsnt in G. Thn, Y(G) = Y(G ) + n DS δ + n DS δ + (n DE(x) δ DE (x)) x B whr B is th st of yl dgs in E and { s s, p =, (p) = 0, ls. r( ) s (p), Lt = n DS δ + n DS δ + x (n DE(x)δ DE (x)). From th prvious stion, w s that 0. Sustituting th xprssion for Y(G ) from (), w gt, Y(G) X + + r( l ) r( ) l Bu(G ) B (s G (p) + s (p)). W assum that th dgs in E ar unoupid in C (G ). Othrwis, thr will two or mor dgr- it nods in G idntifid with th sam dg and d =. Thrfor, r( l ) r( ) = r( l ), l Bu(G ) l Bu(G) B whr B u (G) is th st of unoupid yl-dgs in C (G). Similarly, w assum that th hk nods to whih th nw

13 dgr- singular nods ar onntd to ar unoupid in G to avoid d =. Hn, if s G is th numr of singular nods in G, w hav s G (p) + s (p) = s G (p), f f whr s G (p) = { s G, p = 0, ls. Sustituting th aov xprssions in Y(G), w gt, Y(G) X + + r( l ) s G(p). l Bu(G) From th proof of Lmma w s that, X > 0 for p = (R ). Cas : p Th trm l Bu(G) r( l ) s G(p) is non-ngativ. Sin 0, Y(G) > 0. Cas : p = If l Bu(G) r( l ) s G(p) 0, Y(G) > 0 for p = and thr xists a odword-supporting omplt -graph on (R ) p = = nods. Evn othrwis, y Proposition, thr xists a odword supporting sugraph on nods. F. Girth ondition and final proof W now omplt th proof of th main rsult y showing that th girth ondition holds for suital rats. Proposition : Considr a planar Tannr graph G with minimum distan d >. Lt g th girth of C (G). Thn, g. Hn, th girth ondition is satisfid for p. Proof: It is asy to s that thr annot any loops in C (G). W will show that multipl dgs in C (G) will rsult in d = to prov th proposition. Lt f and two nods in C (G) onntd y two dgs. An dg twn f and in th dual orrsponds to a ommon dg twn th two fas f and in th C(G). Hn, th two dgs onnting f and orrspond to two ommon dgs twn th fas f and. Sin ah fa is of lngth in C(G), oth fas f and hav th sam vrtx st. Suh a situation ariss in th onstrution of C(G) only whn two dgr- it nods in G hav th sam st of nighoring hk nods i.. whn d =. Hn, whn d > thr ar no multipl dgs in C (G). Sin p rsults in h(p), th girth ondition for p nds g, whih is satisfid for C (G). Exampl 7: A situation with multipl dgs is shown in Fig. 5 whr U = {f, }. In Fig. 5, th sugraph C U (G) of th hk graph orrsponding to multipl dgs in C (G) is shown. Th dottd lins show th dgs and nods of G. W s that multipl dgs in C (G) rsult from two dgr- it nods in G having th sam st of nighouring hk nods, in whih as d =. Osrv that p = (R ) 0. Whn 5 8 R < 7 8, w hav p. By Proposition and Lmma, thr Fig. 5. C U (G) C U (G) Multipl dgs in th dual of hk graph. xists a odword-supporting sugraph on By Proposition, 7 8R d (R ) (R ) nods. +. (8) Whn R 7 8, y Proposition, d for th minimum distan d of th od dfind y th hk invrs G. Sin th DS and DE oprations in th onvrsion from G to G annot dras th sum i w i in Proposition, th sam ound holds for th minimum distan of G. So, w hav (R ) d = + for R 7 8. This onluds th proof of Thorm 5, whih is th main rsult of this papr. A plot of th uppr ound of Thorm 5 on d vrsus R for planar ods is shown in Fig. 6. Minimum distan (d) Fig Rat (R) Minimum distan vrsus rat for planar ods. Th girth of C (G) annot aritrarily larg. In fat, all C (G) hav girth g 5. This is aus th girth of C (G) is lssr than or qual to th minimum nod dgr in C(G), whih is lss than 6 y planarity; hn, g an tak a maximum valu of 5. Corollary 5: Lt G a planar Tannr graph whih supports a od of rat R 9 6 with th girth of C (G) gratr than or qual to 5. Thn, d 7 8R (R ) +. (9)

14 Proof: Its asy to s that th girth ondition is mt for all p 0 (sin p =. l(p) + z(p)). With similar alulations as aov, on an show that this plas rstrition on th rat as R 9 6. V. CONCLUSION In this papr, w showd a ound on th minimum distan of high-rat ( 5/8) ods that hav planar Tannr graphs. Th main rsult is th plot of th uppr ound on minimum distan as a funtion of rat as shown in Fig. 6. In partiular, w s that suh ods hav a maximum minimum distan of 7. Hn, non-planarity is ssntial for th onstrution of ods on graphs with high minimum distan. Th proof uss idas from graph thory, oding thory and an avraging argumnt through a sris of onstrutions that xploit th planarity of a Tannr graph. Idas from th proof ould possily mployd in onstrution of ods on nonplanar graphs in th futur. Extnding th ound to ods of all rats with a planar Tannr graph is an intrsting prolm for futur study. W onjtur that ods with planar Tannr graphs will not support ods with larg minimum distan for any rat. APPENDIX A ORDERING NODES IN C U (G) In this appndix, w show that C U (G) for a propr sust of nods U an rursivly onstrutd y adding th nods from U on at a tim in an ordr suh that ah nwly addd nod has dgr or. For an ordrd st U = {u, u, u U }, lt U i = {u, u,, u i }. Proposition 6: Lt C U (G) a onntd sugraph of C (G) indud y a propr nod sust U. Thn, thr xists an ordring of th nods in U, givn y U = {u, u, u U }, suh that th dgr of th nod u i in C U i (G) is ithr or, and C U i (G) is onntd for i U. To prov th proposition, w first laim th following: Claim: Lt C V (G) a sugraph of C (G) indud y a propr sust V of nods. Thn thr xists ithr a dgr- nod in C V (G) or a dgr- nod that is not a ut-vrtx of C V (G). Proof of laim: W s that C V (G) has at last on nod v of dgr, sin C V (G) is a propr sugraph of C (G). If v is a dgr- or a dgr- non-ut vrtx, w ar don. Othrwis, if v is a dgr- ut-vrtx nod, lt V and V th vrtx sts of th two omponnts of C V (G) v. If all th nods in V ar of dgr in C V (G), thn th dg joining v and V will a ut dg in C (G). A ut dg in C (G) implis a loop in C(G) [7], whih is not possil y onstrution. By similar argumnts for V, w s that thr is a nod in V and a nod in V with dgr in C V (G). Lt v a nod in V of dgr in C V (G). If th nod v is of dgr or if it is a dgr- non-ut vrtx, w ar don. Othrwis, prod with v in pla of v and C V (G) in pla of C V (G). Sin th vrtx st is finit, w ar guarantd to find a dgr- or a dgr- non-ut vrtx proding to smallr omponnts. W ar now rady to prov th proposition. Proof: In C U (G), lt u U a dgr- or a dgr- non-ut vrtx. For i U, lt u i a dgr- or a dgr- non-ut vrtx in C U (G) u U u U u i+. Osrv that C U (G) u U u U u i+ is onntd. Th rquird ordring is thn givn y {u, u,, u U }. APPENDIX B COMPUTATION OF η Proposition 7: Lt G th hk invrs of a planar Tannr graph G. Lt η dfind as in (). Thn, η = V ( p + ) l Bo(G ) r( l ) (0) providd th girth ondition is satisfid for C (G). Proof: A dgr-two it nod v is idntifid with an dg in C(G ). Sin is ommon to two fas, say f and, v is ountd whnvr ithr or oth th orrsponding nods (with sam notation f and as in Stion II-D ) in C (G ) is in th vrtx st of omplt -graphs on p vrtis. Lt th dg onnting f and in C (G ) orrsponding to i., th dg that is oupid y v. Th situation is dpitd in Fig. 7. Lt T and T th st of omplt -graphs Fig. 7. Computing η. f Two ranhs Two ranhs rootd at f rootd at ontaining f and, rsptivly. Thn, q(v ) = T T = T + T T T. Sin T = T = p (assuming th girth ondition), w ar lft with omputing T T. Not that T T is th st of omplt -graphs that ontain oth f and i., ontaining as shown in Fig. 7. W will now ount th numr of -trs that will ontain. ) Thr ar two ranhs rootd at f and two at as sn in Fig. 7. Th numr of nods up to lvl l(p) from th rsptiv roots in ths four ranhs is ( l(p) ). Evry omplt -tr rootd at any of ths nods will ontain. ) Th numr of possil nods in lvl l(p) in th four ranhs is ( l(p) ). Among th omplt -trs rootd at ths nods, a fration z(p) will ontain l(p). ) Finally, all omplt -trs rootd at f and will ontain.

15 5 Hn, th total numr of omplt -trs that ontain is givn y ( ) ( l(p) ) + ( l(p) z(p) ) +. l(p) In addition, thr will r( ) omplt -graphs that ontain as a yl-rating dg. Thrfor, T T = ( l(p) + ) z(p) + r( ), Hn, q(v ) for v V is Ltting w hav η = = ( p ) + r( ). () q(v ) = ( p + ) r( ). θ(p) = ( p + ), () v V θ(p) l Bo(G ) = V θ(p) l Bo(G ) r( ) r( l ), f f v (a) Fig. 8. Computing q(v ) for a dgr- nod v. f 5 f f f 6 p Cas from Fig. 8 q(v ) θ(p) (a),() (a),() 0 (a) 5 () (a) 6 6 () 5 6 TABLE II ENUMERATION OF WORST-CASE q(v ). v f 5 () f f onntd to othr nighours. Th valus of q(v ) in ths ass an radily sn to muh largr than θ(p) in a similar fashion. f APPENDIX C COMPUTATION OF η Proposition 8: Lt G a planar Tannr graph. Lt s G th numr of singular nods in G. Thn, providd th girth ondition on C (G) is satisfid, f f η V ( p + ) s G(p), whr η is dfind in () and { s G, p =, s G (p) = 0, ls. Proof: A dgr- it nod v in G is onntd to on hk nod, and is idntifid with that nod in C(G). Evry nod in C(G) maps to a fa in C (G). Lt V (v ) th st of vrtis of C (G) forming th fa orrsponding to N (v ). Thn v ontriuts to th summation of wights whnvr on or mor of th vrtis in V (v ) is in th vrtx st of omplt -graphs. For p, th girth of C (G) is at last. Two possil situations with girth ar shown in Fig. 8. Tal II numrats q(v ) for p =,,, for th two ass in Fig. 8 and ompars with θ(p) = ( p + ). From Tal II, w s that th only as whn q(v ) < θ(p) is for p = in th situation of Cas (). Lt U = {f,, f, f } in Fig. 8. Thn for th sugraph C U (G), th orrsponding hk graph C U (G) is shown in Fig. 9 as dashd lins. Hn, y th dfinition in Stion IV-C, v is singular. For p 5, w hav girth at last 5 aording to th girth ondition. Hn, th fa of C (G) orrsponding to v has at last fiv nods Fig. 9. f C U (G) and C U (G) for singular as. Thrfor, w s that whr η V ( p + ) s G(p), s G (p) = { s G, p =, 0, ls. APPENDIX D COMPUTATION OF α x Proposition 9: Lt G a planar Tannr graph with a fixd mdding. Lt Y(G) dfind as in (9). Lt α x th inras in Y(G) whn a nw it nod of dgr x + is mddd in G suh that th rsulting graph is still planar. Thn, ( α p ),

16 6 and α ( p ) 6 z(p). whr p is suh that girth ondition is satisfid for C (G). Proof: Whn x =, a dgr- it nod v oms a dgr- it nod and v is idntifid with two fas of th hk graph having a ommon dg. This is quivalnt to saying that v is idntifid with an dg in th dual. So, α is qual to th numr of omplt -graphs that ontain th dg. This numr is alrady drivd in (), and w gt α ( p ). Whn x =, a dgr- it nod v oms a dgr-5 it nod as shown in Fig. 0. Th nod v is idntifid with thr fas f, and f of th hk graph C(G). This orrsponds to th onntd sugraph C U (G) with U = {f,, f } in th dual as shown in Fig. 0. So, α is qual to th numr of REFERENCES [] R. G. Gallagr, Low-Dnsity Parity-Chk Cods. Camridg, MA: MIT Prss, 96. [] R. M. Tannr, A rursiv approah to low omplxity ods, IEEE Trans. on Info. Thory, vol. 7, no. 5, pp. 5 57, Sptmr 98. [] T. Etzion, A. Trahtnrg, and A. Vardy, Whih ods hav yl-fr tannr graphs? IEEE Trans. on Info. thory, vol. 5, no. 6, pp. 7 8, Sp 999. [] S. Srimathy and A. Thangaraj, Cods that hav tannr graphs with nonovrlapping yls, in 5th Intrnational Symposium on Turo Cods and Rlatd Topis, Sp 008, pp [5] R. M. Tannr, Minimum-distan ounds y graph analysis, IEEE Trans. on Info. thory, vol. 7, no., pp , F 0. [6] J. A. Bondy and U. S. R. Murty, Graph Thory With Appliations. North- Holland, 976. [7] F. Harary, Graph Thory. Addison-Wsly Pulishrs, 969. f f Branhs rootd at f, f f C U (G) f C U (G) Fig. 0. Computation of α. omplt -graphs whih ontain C U (G). Lt us now omput α. All th omplt -graphs rootd at nods at a dpth of l(p) or lssr from th nod in th ranhs shown in Fig. 0 ontain C U (G). Sin girth ondition is satisfid, thr ar (. l(p) ) suh omplt -graphs. Now, onsidr only th hild nods of f and f in th ranhs shown in Fig. 0 that ar at a dpth of l(p) from. Sin girth ondition is satisfid, th numr of suh nods is l(p) z(p) and thr ar ralizations of omplt -graphs l(p) rootd at ah suh nod ontaining C U (G). Thus, ( α. l(p) +. l(p) z(p). ( l(p) p ) 6 z(p). ),