On the Power of BFS to Determine a Graphs Diameter

Transcription

1 On the Power of BFS to Determine Grphs Dimeter (Etene Astrt) Derek G Corneil 1, Feoor F Drgn 2, n Ekkehr Köhler 3 1 Dept of Compter Siene, Uniersity of Toronto, Toronto, Ontrio, Cn, g@storontoe 2 Dept of Compter Siene, Kent Stte Uniersity, Kent, Ohio, USA, rgn@skente 3 Fhereih Mthemtik, Tehnishe Uniersität Berlin, Berlin, Germny, ekoehler@mthtu-berlinde Astrt Reently onsierle effort hs een spent on showing tht eiogrphi Breth First Serh (BFS) n e se to etermine tight on on the imeter of grphs from rios restrite lsses In this pper, we show tht in some ses, the fll power of BFS is not reqire n tht other ritions of Breth First Serh (BFS) sffie The restrite grph lsses tht re menle to this pproh ll he smll onstnt pper on on the mimm size yle tht my pper s n ine sgrph We show tht on grphs tht he no ine yle of size greter thn k, BFS fins n estimte of the imeter tht is no worse thn im(g) k/2 2 1 Introtion Reently onsierle ttention hs een gien to the prolem of eeloping fst n simple lgorithms for rios lssil grph prolems The motition for sh lgorithms stems from or nee to sole these prolems on ery lrge inpt grphs, ths the lgorithms mst e not only fst, t lso esily implementle Determining the imeter of grph is fnmentl n seemingly qite time onsming opertion For ritrry grphs (with n erties n m eges), the rrent fstest lgorithm rns in O(nm) time whih is too slow to e prtil for ery lrge grphs This nie lgorithm emines eh erte in trn n performs Breth First Serh (BFS) strting t the prtilr erte Sh sweep strting t erte immeitely etermines e(), the eentriity of erte Rell tht the eentriity of erte, e() =m y V (, y), where (, y) enotes the istne etween n y; theimeter of G eqls the mimm eentriity of ny erte in V It is ler tht this lgorithm tlly omptes the entire istne mtri; lerly knowing the istne mtri immeitely yiels the imeter of the grph For ense grphs, the est reslt known is y Seiel [12], who showe tht the ll pirs shortest pth prolem (n hene the imeter prolem) n e sole S Rjsm (E): ATIN 2002, NCS 2286, pp , 2002 Springer-Verlg Berlin Heielerg 2002

2 210 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler in O(M(n)logn) timewherem(n) enotes the time ompleity for fst mtri mltiplition inoling smll integers only The rrent est mtri mltiplition lgorithm is e to Coppersmith n Winogr [2] n hs O(n 2376 )time on Unfortntely, fst mtri mltiplition lgorithms re fr from eing prtil n sffer from lrge hien onstnts in the rnning time on Note tht no effiient lgorithm for the imeter prolem in generl grphs, oiing the ompttion of the whole istne mtri, hs een esigne Ths, the qestion whether for grph the imeter n e ompte esier thn the whole istne mtri remins still open Clerly, performing BFS strting t erte of mimm eentriity esily proes the grph s imeter Ths one wy to pproimte the imeter of grph is to fin erte of high eentriity; this is the pproh tken in this pper This is not, howeer, the only pproh For emple, Aingworth et l [1] otin rtio 2/3 pproimtion to the imeter in time O(m nlogn+n 2 logn) Note tht rtio of 1/2 n esily e hiee y hoosing n ritrry erte (the eentriity of ny erte is t lest one hlf the imeter of the grph) n performing BFS strting t this erte It follows lso from reslts in [1,6] (see lso pper [13] whih sreys reent reslts relte to the ompttion of et n pproimte istnes in grphs) tht the imeter prolem in nweighte, nirete grphs n e sole in Õ(min{n3/2 m 1/2,n 7/3 }) time with n itie error of t most 2 withot mtri mltiplition Here, Õ(f) mens O(f polylog(n)) The motition ehin the work of Aingworth et l is to fin fst, esily implementle lgorithm (they oi sing mtri mltiplition), motition tht we shre Or pproh is to emine the nie lgorithm of hoosing erte, performing some ersion of BFS from this erte n then showing nontriil on on the eentriity of the lst erte isite in this serh In ft, this lgorithm is one of the lssil lgorithms in grph theory; if one restrits one s ttention to trees, then this lgorithm proes erte of mimm eentriity (see eg [9]) This pproh hs lrey reeie onsierle ttention (In the following we let enote the erte tht ppers lst in prtilr serh; the efinition of the rios serhes n fmilies of grphs will e presente in the net setion) For emple, Drgn et l [8] he shown tht if BFS is se for horl grphs, then e() im(g) 1wheresforinterl grphs e() =im(g) It is ler from the work of Corneil et l [4], tht y sing BFS on AT-free grphs, one hs e() im(g) 1 Drgn [7], gin sing BFS, hs shown tht e() im(g) 2 for HH-free grphs, e() im(g) 1 for HHD-free grphs n e() =im(g) for grphs tht re oth HHD-free n AT-free It is interesting to note tht Corneil et l [4] he looke t ole sweep BFSs (ie strt n BFS from erte tht is lst in preios ritrrily hosen BFS) on horl n AT-free grphs They he proie forien sgrph strtre on grphs where e() =im(g) 1 They lso presente oth horl n AT-free grphs where for no, isthe-sweep BFS lgorithm grntee to fin erte of mimm eentriity Frthermore, they showe

3 On the Power of BFS to Determine Grphs Dimeter 211 tht for ny, there is grph G where e() im(g), where is the erte isite lst in 2-sweep BFS This grph G, howeer, hs lrge ine yle whose size epens on These reslts motite nmer of interesting qestions: Is it n inherent property of BFS to en in erte of high eentriity for the rios restrite grph fmilies mentione oe? Wht hppens if we se other rints of BFS? Why o AT-free n horl grphs, two fmilies with ery isprte strtre, ehiit sh similr ehior with respet to the effiy of BFS to fin erties of high eentriity? Althogh BFS fils to fin erties of high eentriity for grphs in generl, ll known emples tht ehiit sh filre he lrge ine yles If we on the size of the lrgest ine yle, n we get on on the eentriity of the erte tht ppers lst in n BFS? If the preios qestion is nswere in the ffirmtie, is the fll power of BFS neee? Wht hppens if we jst se BFS? This pper resses these qestions In the net setion, we present the rios forms of BFS n efine the grph theoreti terminology se throghot the pper In Setion 3 we emine the ehior of the ifferent ersions of BFS on rios restrite grph fmilies We estlish some new ons n show, y emple, tht ll stte ons on e() re tight In Setion 4, we emine fmilies of grphs where the size of the lrgest ine yle is one n show tht BFS oes see in getting erties of high, with respet to k, eentriity 2 Nottion n Definitions First we formlize the notion of BFS n then isss rios ritions of it We tion the reer tht there is some onfsion in the litertre etween BFS n wht we ll, efine elow In efining the rios ersions of BFS, we re only onerne with ientifying the lst erte isite y the serh; strightforwr moifitions proe the list of erties in the orer tht they re isite y the serh It shol e note tht none of the orerings re niqe; inste, eh serh ientifies one of the possile en-erties Algorithm BFS: Breth First Serh Inpt: grph G(V,E) nerte Otpt: erte, the lst erte isite y BFS strting t Initilize qee Q to e {} n mrk s isite while Q o et e the first erte of Q n remoe it from Q Eh nisite neighor of is e to the en of Q n mrke s isite

4 212 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler Note tht the oe lgorithm n esily e moifie to otin the lyers of V with respet to In prtilr, for eh 0 i e(), the i-th lyer of V with respet to, enote i (), = { :(, ) =i} This motites the net lgorithm, Algorithm : st yer Inpt: grph G(V,E) nerte Otpt: erte, erte in the lst lyer of Rn BFS to get the lyering of V with respet to Choose to e n ritrry erte in the lst lyer Clerly ny erte retrne y BFS n lso e retrne y ; the onerse is not tre s shown y the grph in Figre 1 e Fig 1 No BFS strting t n retrn erte Now we moify this lgorithm to otin erte in the lst lyer tht hs minimm egree with respet to the erties in the preios lyer Algorithm +: st yer, Minimm Degree Inpt: grph G(V,E) nerte Otpt: erte, erte in the lst lyer of, tht hs minimm egree with respet to the erties in the preios lyer Rn BFS to get the lyering of V with respet to Choose to e n ritrry erte in the lst lyer tht hs minimm egree with respet to the erties in the preios lyer Finlly we introe eiogrphi Breth First Serh (BFS) This serh prigm ws isoere y Rose et l [11] n ws shown to yiel simple liner time lgorithm for the reognition of horl grphs In light of the gret el

5 On the Power of BFS to Determine Grphs Dimeter 213 of work rrently eing one on BFS, it is somewht srprising tht interest in BFS ly ormnt for qite while fter [11] ppere Algorithm BFS: eiogrphi Breth First Serh Inpt: grph G(V,E) nerte Otpt: erte, the lst erte isite y n BFS strting t Assign lel to eh erte in V for i = n ownto 1 o Pik n nmrke erte with the lrgest (with respet to leiogrphi orer) lel Mrk isite For eh nmrke neighor y of, i to the lel of y If erte e is remoe from the grph in Figre 1, we he grph where erte, nmely, n e isite lst y BFS from t y no BFS from We now trn to the efinitions of the rios grph fmilies introe in the preios setion A grph is horl if it hs no ine yle of size greter thn 3 An interl grph is the intersetion grph of interls of line ekkerkerker n Boln [10] efine n steroil triple to e triple of erties sh tht etween ny two there is pth tht ois the neighorhoo of the thir n showe tht grph is n interl grph iff it is oth horl n steroil triple-free (AT-free) A lw is the omplete iprtite grph K 1,3,hole is n ine yle of length greter thn 4, hose is 4-yle with tringle e to one of the eges of the C 4 n omino is pir of C 4 s shring n ege A grph is HH-free if it ontins no ine hoses or holes n is HHD-free if it ontins no ine hoses, holes or ominos Finlly, in orer to ptre the notion of smll ine yles, we efine grphtoek-horl if it hs no ine yles of size greter thn k Note tht horl grphs re preisely the 3-horl grphs n AT-free grphs re 5-horl We efine the isk of ris r entere t to e the set of erties of istne t most r to, ie, D r () ={ V :(, ) r} = r i=0 i() 3 Restrite Fmilies of Grphs We now see how the for serh lgorithms mentione in the preios setion ehe on the following fmilies of grphs: horl, AT-free, {AT,lw}-free, interl, n hole-free The reslts re smmrize in the following tle In this tle the referenes refer to the pper where the lower on ws estlishe; [*] inites tht the reslt is new A figre referene refers to the pproprite figre where it is shown tht the lower on is tight In eh of the figres the erte pir, forms imetrl pir, ie (, ) =im(g) Below eh figre BFS, BFS,, or + orering is gien tht hiees the orresponing ons; erte is lwys the strt-erte n the en-erte of the pproprite serh; ifferent BFS-lyers re seprte y

6 214 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler GRAPH CASS + BFS BFS D 2 D 2 D 1 D 1 horl grphs [3] [3] [*] [8] Fig 4 Fig 5 Fig 2 Fig6 D 2 D 1 D 2 D 1 AT-free grphs [*] [*] [*] [4] Fig 3 Fig 7 Fig 3 Fig 7 D 1 = D D 1 = D {AT,lw}-free grphs [*] [*] [*] [*] Fig 2 Fig 2 D 1 = D D 1 = D interl grphs [*] [*] [*] [8] Fig 2 Fig 2 D 2 D 2 D 2 D 2 hole-free grphs [*] [*] [*] [*] Fig 8 Fig 8 Fig 8 Fig 8 f g e f e g h Fig 2 BFS: Fig 3 BFS: ge f Fig 4 : gh ef e g e i f f i k h g h Fig 5 ik fegh +: Fig 6 gh ef i BFS: To illstrte the types of tehniqes tht re se to estlish lower ons in the tle, we now show tht e() im(g) 1 for horl grphs when BFS is se This reslt ssmes the reslt shown in [8] tht this lower on hols when BFS is se The jornl ersion of the pper will ontin proofs for ll new reslts mentione in the tle

7 On the Power of BFS to Determine Grphs Dimeter 215 i e f h g Fig 7 BFS: Fig 8 BFS: ghi fe First we omment on the BFS lgorithm In prtilr, we my regr BFS s hing proe nmering from n to 1 in eresing orer of the erties in V where erte is nmere n As erte is ple on the qee, it is gien the net ille nmer The lst erte isite,, is gien the nmer 1 Ths BFS my e seen to generte roote tree T with erte s the root Aertey is the fther in T of etly those neighors in G whih re inserte into the qee when y is remoe An orering σ= [ 1, 2,, n ] of the erte set of grph G generte y BFS will e lle BFS orering of G etσ(y) e the nmer ssigne to ertey in this BFS orering σ Denotelsoyf(y) the fther of erte y with respet to σ The following properties of BFS orering will e se in wht follows Sine ll lyers of V onsiere here re with respet to, we will freqently se nottion i inste of i () (P1) If y q (q>0) then f(y) q 1 n f(y)istheertefromn(y) q 1 with the lrgest nmer in σ (P2) If i, y j n i<j,thenσ() >σ(y) (P3) If, y j n σ() >σ(y), then either σ(f()) >σ(f(y)) or f() =f(y) (P4) If, y, z j, σ() >σ(y) >σ(z)nf()z E,thenf() =f(y) =f(z) (in prtilr, f()y E) In wht follows, y P (, y) we will enote pth onneting erties n y Proof of the following lemm is omitte emm 1 If erties n of isk D r () of horl grph re onnete y pth P (, ) otsie of D r () (ie, P (, ) D r () ={, }), then n mst e jent et σ e BFS orering of horl grph G strte t erte et lso P (, ) =( = 1, 2,, k 1, k = ) e shortest pth of G onneting erties n WesythtP (, ) isrightmost shortest pth if the sm σ( 1 )+ σ( 2 )++ σ( k 1 )+ σ( k ) is the lrgest mong ll shortest pths onneting n

8 216 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler emm 2 et n y e two ritrry erties of horl grph G Eery rightmost shortest pth P (, y) etween n y n e eompose (see Figre 9) into three shortest spths P =( = 1, 2,, l ) (lle the ertil spth), P y =(y = y 1,y 2,,y k ) (lle the ertil y spth) n P ( l,y k ) (lle the horizontl spth) sh tht 1 P ( l,y k ) j for some j {0, 1,,e()}, n( l,y k ) 2 (ie, l n y k either oinie or re jent or he ommon neighor in G[ j ] P ( l,y k )); 2 l i j+i for 0 i l 1; 3 y k i j+i for 0 i k 1 y = y 1 y 2 y = y 1 y 2 = 1 y 2 y = y 1 = 1 2 y k-1 or = 1 2 l-1 y k-1 or 2 l-1 y k-1 j j-1 l-1 l y k j j-1 l y k y k = l j j Fig 9 The strtre of rightmost shortest pth in horl grphs Proof First we proe tht P (, y) i 3 for ny i =1, 2,,e(s) Assme tht the intersetion of P (, y) nlyer q, for some ine q, ontinstlest for erties et,,, e the first for erties of P (, y) q on the wy from to y We lim tht,, E If / E then, y emm 1, spth P (, ) of the pth P (, y) is ompletely ontine in isk D q (s) In prtilr, the neighor of on P (, ) elongs to D q 1 (s) Using the sme rgments, we onle tht E or spth P (, ) of the pth P (, y) is ontine in D q (s) If E then neighor of in q 1 mst e jent with (y emm 1) Sine σ() >σ(), we get ontrition to P (, y) is rightmost pth (we n reple erte of P (, y) with n get shortest pth etween n y with lrger sm) If / E then the neighor of on P (, ) is lso ontine in D q 1 (s) By emm 1, erties n mst e jent, t this is impossile sine P (, y) is shortest pth

9 On the Power of BFS to Determine Grphs Dimeter 217 Ths, erties n he to e jent If erties n re not jent, then the neighor of on P (, ) elongs to D q 1 (s) n, y emm 1, it mst e jent to ny neighor of in q 1 Sineσ() >σ() hols, gin ontrition to P (, y) is rightmost pth rises Conseqently, erties n re jent, too Completely in the sme wy one n show tht n he to e jent Note lso tht the jeny of with n with is proe withot sing the eistene of the erte We he now,, E n the ine pth (,,, ) is rightmost shortest pth (s spth of the rightmost shortest pth P (, y)) Consier neighors n in q 1 of n, respetiely Sine σ( )+ σ( ) > σ()+ σ() nthepth(,,, ) is rightmost n shortest, erties n n neither oinie nor e jent Bt then we get pth (,,,,, ) onneting two non-jent erties of D q 1 (s) otsie of the isk A ontrition to emm 1 otine shows tht P (, y) i 3holsforny i =1, 2,,e(s) Now let P (, y) q =3nlet,, e the erties of P (, y) q on the wy from to y It follows from the isssion oe tht, E n oth the neighor of on spth P (, ) ntheneighor of on spth P (, y) (if they eist) elong to the lyer q+1 If for emple q 1, then, y emm 1, mst e jent to ny neighor of in q 1 Agin, sine σ() >σ(), ontrition to P (, y) is rightmost pth rises Frthermore, if for some ine q, P (, y) q = {, }, n P (, y) = (,,,,,,,y), then oth n elong to the lyer q+1 Smmrizing ll these we onle tht, while moing from to y long the pth P (, y), we n he only one horizontl ege or only one pir of onsetie horizontl eges Here y horizontl ege we men n ege with oth en-erties from the sme lyer All other erties of the pth P (, y) elong to higher lyers Hing the strtre of rightmost shortest pth estlishe, we n now proe the min reslt for horl grphs In presenting rightmost shortest pth, we se / s to ifferentite the pproprite spths Theorem 1 et e the erte of horl grph G lst isite y BFS Then e() im(g) 1 Proof et, y e pir of erties sh tht (, y) =im(g), n onsier two rightmost shortest pths P (, ) =( = 1, 2,, l 1 / l,, h/ h 1,, 2, 1 = ) n P (y, ) =(y = y 1,y 2,,y k 1 /y k,, g / g 1,, 2, 1 = ) onneting erte with n y, respetiely (see Figre 10) By emm 2, eh of these pths onsists of two (perhps of length 0) ertil spths n one horizontl pth of length not greter thn 2 Assme, withot loss of generlity, tht h g n let l,, h qsine e() we lso he l h n k g

10 218 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler By emm 1, erties h, h in q either oinie or re jent Note tht, if ( l, h ) 1, then (, y) (, l)+1+( h,y) (, h )+1+( h,y)= (, y)+1 e() + 1 Tht is, e() (, y) 1=im(G) 1, n we re one Hene, we my ssme tht ( l, h ) 2 n, therefore, l h e() y q q-1 l l h " h " h+1 " g y k Fig 10 Rightmost shortest pths P (, ) np (y, ) We istingish etween two ses The first one is simple Only for the seon se we will nee to se the speil properties of BFS orering Cse g>h In this se there eists erte h+1 in the intersetion P (y, ) q 1 Consier lso neighor l of l in q 1 Sine erties l n h+1 re onnete y pth ( l, l,, h, h, h+1 ) otsie of the isk D q 1(), y emm 1, they re jent if they o not oinie Hene, ( l, h+1 ) 2 n, therefore, (, y) (, l )+2+( h+1,y) (, h )+2+( h+1,y)=(, y)+1 e()+1, ie, gin e() (, y) 1=im(G) 1 Cse g = h From the isssion oe (now, sine g = h we he symmetry), we my ssme tht (y k, h ) 2ny k h Consier neighors l n y k in q 1 of erties l n y k, respetiely (see Figre 11()) By emm 1, they re jent if o not oinie, ie, ( l,y k ) 3 Now, if t lest one of the eqlities ( l, h )=2,(y k, h ) = 2 hols, then we re one Inee, if for emple ( l, h ) = 2, then (, ) = (, l)+2+( h,) n, therefore, (, y) (, l )+( l,y k )+(y k,y) (, l )+3+( h,)=(, )+1 e()+1

11 On the Power of BFS to Determine Grphs Dimeter 219 e() y y l h " h y k q k-1 k k " k-1 " k y k l y k q-1 g f () () Fig 11 () Horizontl spths of P (, ) np (y, ) reinthesmelyer, () Pths P (, ) np (y, ) he similr shpe n the sme length 2k 1 So, we my ssme tht l h,y k h E Moreoer,sine(y k, h ) 2, erties h n h nnot oinie, ie, they re jent If l<hor k<h or ( l,y k ) < 3, gin we will get (, y) e() + 1 y ompring istnes (, y) =h + k 1, (, ) =h + l 1with(, y) l 1+( l,y k )+k 1 l+1+k Ths, we rrie t sittion when l = k = h, l h,y k h, h h E, n ( l,y k )=3,(, ) =(, y) =2k 1 (see Figre 11()) We my lso ssme tht (, y) =2k + 1, sine otherwise, (, y) 2k =(, y) +1 e() +1 n we re one We show tht this finl onfigrtion (with (, y) =2k +1) is impossile ese of the properties of BFS orerings Assme, withot loss of generlity, tht σ(y k ) >σ( k ) n onsier the fthers f = f(y k ), g = f( k )ofy k n k, respetiely Sine ( k,y k )=3,we he f g n f k,gy k / E By emm 1 n property (P3) of BFS orering, erties f n g re jent n σ(f) >σ(g) Chorl grphs nnot ontin n ine yle of length greter thn 3 Therefore, in the yle forme y g, k, k, k,y k,f,tlesthorsg k n f k mst e present Sine for the fther f( k )of k we he σ(f( k )) σ(f) >σ(g), ineqlity σ( k ) >σ( k) mst hol (here we se properties (P1) n (P3) of BFS orerings) We will nee the ineqlity σ( k ) >σ( k) lter to get or finl ontrition Now onsier erties k 1 n k 1 We lim tht σ( k 1 ) <σ( k 1) Assme tht this is not the se, n let j (j {1, 2,,k 2}) ethelrgest ine sh tht σ( j ) <σ( j) (rell tht σ( 1 )=σ() =1<σ() =σ( 1)) Then σ( j+1 ) >σ( j+1) hols, n sine j k 2 n (, ) =2k 1, we otin ( j, j ) 5 (ese of 2k 1=(, ) (, j )+( j, j)+( j,)=

12 220 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler 2(j 1)+( j, j) 2(k 3)+( j, j)=2k 6+( j, j)) Consier the fther t = f( j )of j From the istne reqirement n properties of BFS orerings we onle tht t j+1, t j+1 n σ(t) >σ( j+1 ) >σ( j+1) Moreoer, erte t hs to e jent to j+1 (y emm 1), t nnot e jent to j+2 (sine σ(t) >σ( j+1 )nthepthp(, ) is rightmost) Consier now the fther z = f(t) oftheertet Itisjentto j+2, y emm 1, n hs to e jent to j+1, to oi n ine yle (z,t, j+1, j+2,z)oflength 4 Applying property (P4) to σ(t) >σ( j+1 ) >σ( j+1) nz j+1 E, weget z j+1 E whih is impossile sine ( j, j ) 5 This ontrition shows tht, inee, the ineqlity σ( k 1 ) <σ( k 1) msthol So, we he σ( k 1 ) <σ( k 1) nσ( k ) >σ( k) We repet or rgments from the preios prgrph onsiering ine k 1 inste of j (Theonly ifferene is tht now we o not he the erte j+2 = k+1 on the pth P (, )) Agin, onsier the fther t = f( k 1 )of k 1 Clerly, t k Sine ( k,y k )=3,ertet is not jent to y k Frthermoret oes not oinie with k sine emm 1 wol reqire k k E Hene, t k n, y property (P3), σ(t) >σ( k ) >σ( k) Verte t mst e jent to k, y emm 1, t nnot e jent to k,sinethepthp(, ) isrightmostnσ(t) >σ( k) To oi n ine yle of length 4, erte t is not jent to k s well q+1 q q-1 t k-1 k z " k-1 k " y k k f Fig 12 Illstrtion to the proof of Theorem 1 Consier lso the fther z = f(t) oftheertet (see Figre 12) If zy k E then (, y) (y, y k )+1+(z, k 1 )+( k 1,) k k 2=2k, n ontrition to the ssmption (, y) =2k+1rises Therefore, z n y k re not jent n hene z f (rell tht f is the fther of y k n f k E, f k / E ) By emm 1, zf E Sineσ(t) >σ( k )nf k E, y properties (P3) n (P1), we get σ(z) σ(f( k )) σ(f), ie σ(z) >σ(f) Conseqently, σ(t) >σ(y k ) Now, erte z nnot e jent to k, sine this wol pply the jeny of z with y k,too(yσ(t) >σ(y k ) >σ( k ) n property (P4)) Bt then, in the yle (z,t, k, k, k,f,z) only hors z k, z k, f k, ft re possile, whih re not enogh to oi n ine yle of length greter thn 3 in G A ontrition with the horlity of G ompletes the proof of the theorem

13 4 k-horl Grphs On the Power of BFS to Determine Grphs Dimeter 221 As mentione in the introtion, the emples tht show tht BFS fils to fin erties of high eentriity ll he lrge ine yles Frthermore, oth horl n AT-free grphs he onstnt ons on the mimm size of ine yles Ths one wol hope tht for k-horl grphs where k is onstnt, some form of BFS wol see in fining erte whose eentriity is within some fntion of k of the imeter In ft, we show tht is sffiiently strong to ensre this First, lemm tht is se in the proof emm 3 If erties n of isk D r () of k-horl grph re onnete y pth P (, ) otsie of D r () (ie, P (, ) D r () ={, }), then(, ) k/2 Proof Assme (, ) > k/2, nletp e n ine spth of P (, ) onneting erties n Consier shortest pths P (, ) n P (, ) (onneting with n with, respetiely) Using erties of these pths we n onstrt n ine pth Q(, ) with the property tht ll its erties eept n re ontine in D r 1 () By or onstrtion, the yle C otine y the ontention of P n Q(, ) is ine Sine (, ) > k/2, oth pths P n Q(, ) mst e of length greter thn k/2 Therefore, the yle C hs the length t lest k/2 +1+ k/2 +1>k, tht is impossile Theorem 2 et G e k-horl grph Consier, strting in some erte of G n let e erte of the lst BFS lyer Then e() im(g) k/2 2 Proof et, y e pir of erties sh tht (, y) =im(g), n onsier two shortest pths P (, ) np (y, ) onneting erte with n y, respetiely et lso q e the minimm ine sh tht q (P (, ) P (y, )) Consier erte z q (P (, ) P (y, )), n ssme, withot loss of generlity, tht z elongs to P (y, ), ie, we he (, z)+(z,y)=(, y) Assme, for now, tht z Consier lso shortest pth Q(, ) etween n, erties q Q(, ), q 1 Q(, ), n neighor w of z in q 1 Sine erties n w elong to D q 1 () n n e onnete otsie of D q 1 (), y emm 3, (,w) k/2 mst hol We he lso (, ) (, z) ese e() Therefore, (, y) (, )+1+(,w)+1+ (z,y) (, z)+ k/2 +2+(z,y)=(, y)+ k/2 +2 e()+ k/2 +2, ie, e() im(g) k/2 2 Finlly, if z =, then similr rgment s oe estlishes tighter on on e() This reslt n e strengthene frther for 4-horl grphs n 5-horl grphs We n proe the following

14 222 Derek G Corneil, Feoor F Drgn, n Ekkehr Köhler Theorem 3 et G e 5-horl grph Consier, strting in some erte of G n let e erte of the lst BFS lyer Then e() im(g) 2 Agin this on on e() is tight Figre 8 represents 4-horl grph G for whih BFS eists sh tht the erte, lst isite y this BFS, hs eentriity eql to im(g) 2 Emples of k-horl grphs with lrger ifferene etween imeter n the eentriity of the erte lst isite in some rint of BFS re gien in the following two figres Figre 13 shows 6-horl grph For this grph G, there is n, strting in erte, tht ens in erte, where e() =im(g) 3 This shows tht the on of im(g) 2 whih hols for 4- n 5-horlgrphs (for n +) n not hol for 6-horl grphs s well In Figre 14 eh of the she eges stns for pth of length k Ths this grph G is 4k-horl The imeter of G is lso 4k, the eentriity of, the erte lst isite y some BFS strte in, is e() =2k + 1 Hene the ifferene etween the imeter n the eentriity e() is2k 1 This shows tht, t lest for the 4k-horl grphs, the on on e() (for)gienin Theorem 2 is lose to the est possile It is within 3 of the on tht ol e hiee y BFS l e n g m f i Fig 13 ik fh lengm k h : Fig 14 BFS: Aknowlegements: DGC n EK wish to thnk the Ntrl Siene n Engineering Reserh Conil of Cn for their finnil spport Referenes 1 D Aingworth, C Chekri, P Inyk n R Motwni, Fst estimtion of imeter n shortest pths (withot mtri mltiplition), SIAM J on Compting, 28 (1999),

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