Disovring Pirwis Comptiility Grphs Muhmm Nur Ynhon, M. Shmsuzzoh Byzi, n M. Siur Rhmn Dprtmnt of Computr Sin n Enginring Bnglsh Univrsity of Enginring n Thnology nur.ynhon@gmil.om, shms.yzi@gmil.om, siurrhmn@s.ut.. Astrt. Lt T n g wight tr, lt T (u, v) th sum of th wights of th gs on th pth from u to v in T,nlt min n mx two nonngtiv rl numrs suh tht min mx. Thn pirwis omptiility grph of T for min n mx is grph G =(V,E), whr h vrtx u V orrspons to lf u of T n thr is n g (u,v ) E if n only if min T (u, v) mx. AgrphG is ll pirwis omptiility grph (PCG) if thr xists n g wight tr T n two non-ngtiv rl numrs min n mx suh tht G is pirwis omptiility grph of T for min n mx. Krny t l. onjtur tht vry grph is PCG [3]. In this ppr, w rfut th onjtur y showing tht not ll grphs r PCGs. W lso show tht th wll known tr powr grphs n som of thir xtnsions r PCGs. Introution Lt T n g wight tr n lt min n mx two non-ngtiv rl numrs suh tht min mx.apirwis omptiility grph of T for min n mx is grphg =(V,E), whr h vrtx u V rprsnts lf u of T n thr is n g (u,v ) E if n only if th istn twn u n v in T lis within th rng from min to mx. T is ll th pirwis omptiility tr of G. W not pirwis omptiility grph of T for min n mx y PCG(T, min, mx ).Agrph G is pirwis omptiility grph (PCG) if thr xists n g wight tr T n two non-ngtiv rl numrs min n mx suh tht G = PCG(T, min, mx ). Figur () pits n g wight tr T n Fig. () pits pirwis omptiility grph G of T for min =4n mx =7; thr is n g twn n in G sin in T th istn twn n is six, ut G os not ontin th g (, ) sin in T th istn twn n is ight, whih is lrgr thn svn. It is quit pprnt tht singl g wight tr my hv mny pirwis omptiility grphs for iffrnt vlus of min n mx. Likwis, singl pirwis omptiility grph my hv mny trs of iffrnt topologis s its pirwis omptiility trs. For xmpl, th grph in Fig. () is PCGofthtrinFig.()for min =4n mx =7, n it is lso PCGof th tr in Fig. () for min =5n mx =6. In th rlm of pirwis omptiility grphs, two funmntl prolms r th tr onstrution prolm n th pirwis omptiility grph rognition prolm. Givn PCG G, th tr onstrution prolm sks to onstrut n g wight tr T, suh tht G is pirwis omptiility grph of T for suitl min n mx.th M.T. Thi n S. Shni (Es.): COCOON 200, LNCS 696, pp. 399 408, 200. Springr-Vrlg Brlin Hilrg 200
400 M.N. Ynhon, M.S. Byzi, n M.S. Rhmn 2 3 2 3 () () () Fig.. () An g wight tr T, () pirwis omptiility grph G, n () n g wight tr T 2 pirwis omptiility grph rognition prolm sks th nswr whthr or not givn grph is PCG. Pirwis omptiility grphs hv thir origin in Phylogntis, whih is rnh of omputtionl iology tht onrns with ronstruting volutionry rltionships mong orgnisms [2,4]. Phylognti rltionships r ommonly rprsnt s trs known s th phylognti trs. From prolm of ollting lf smpls from lrg phylognti trs, Krny t l. introu th onpt of pirwis omptiility grphs [3]. As thir origin suggsts, ths grphs n us in ronstrution of volutionry rltionships. Howvr, thir most intriguing potntil lis in solving th Cliqu Prolm. A liqu in grph G is st of pirwis jnt vrtis. Th liqu prolm sks to trmin whthr grph ontins liqu of t lst givn siz k. It is wll known NP-omplt prolm. Th orrsponing optimiztion prolm, th mximum liqu prolm, sks to fin th lrgst liqu in grph []. Krny t l. hv shown tht for pirwis omptiility grph G, th liqu prolm is quivlnt to lf smpling prolm whih is solvl in polynomil tim in ny pirwis omptiility tr T of G [3]. Sin thir inption, pirwis omptiility grphs hv ris svrl intrsting prolms, n hithrto most of ths prolms hv rmin unsolv. Among th othrs, intifying iffrnt grph lsss s pirwis omptiility grphs is n importnt onrn. Although ovrlpping of pirwis omptiility grphs with mny wll known grph lsss lik horl grphs n omplt grphs is quit pprnt; slight progrsss hv n m on stlishing onrt rltionships twn pirwis omptiility grphs n othr known grph lsss. Phillips hs shown tht vry grph of fiv vrtis or lss is PCG [6] n Ynhon t l. hv shown tht ll yls, yls with singl hor, n tus grphs r PCGs [7]. Sing th
Disovring Pirwis Comptiility Grphs 40 xponntilly inrsing numr of possil tr topologis for lrg grphs, th proponnts of PCGs oniv tht ll unirt grphs r PCGs [3]. In this ppr, w rfut th onjtur y showing tht not ll grphs r PCGs. Pirwis omptiility grphs hv striking similrity, in thir unrlying onpt, with th wll stui grph roots n powrs. A grph G =(V,E ) is k-root of grph G =(V,E) if V = V n thr is n g (u, v) E if n only if th lngth of th shortst pth from u to v in G is t most k. G is ll th k-powr of G [5]. A spil s of grph powr is th tr powr, whih rquirs G to tr. Tr powr grphs n thir xtnsions (Stinr k-powr grphs, phylognti k-powr grphs, t.) r y finition similr to pirwis omptiility grphs. Howvr, th xt rltionship of ths grph lsss with pirwis omptiility grphs ws unknown. In this ppr, w invstigt th possiility of th xistn of suh rltionship, n show tht tr powr grphs n som of thir xtnsions r in ft pirwis omptiility grphs. Suh rltionship my srv th purpos of not only unifying rlt grph lsss ut lso utilizing th mtho of tr onstrutions for on grph lss in nothr. Th rst of th ppr is orgniz s follows. Stion 2 sris som of th finitions w hv us in our ppr, Stion 3 shows tht not ll grphs r pirwis omptiility grphs. Stion 4 stlishs rltionship of tr powr grphs n thir xtnsions with pirwis omptiility grphs. Finlly, Stion 5 onlus our ppr with isussions. 2 Prliminris In this stion w fin som trms tht w hv us in this ppr. Lt G =(V,E) simpl grph with vrtx st V n g st E. Th sts of vrtis n gs of G r not y V (G) n E(G), rsptivly. An g twn two vrtis u n v of G is not y (u, v). Two vrtis u n v r jnt n ll nighors if (u, v) E; th g (u, v) is thn si to inint to vrtis u n v. Thgr of vrtx v in G is th numr of gs inint to it. A sugrph of grph G =(V,E) is grph G =(V,E ) suh tht V V n E E; w thn writ G G. IfG ontins ll th gs of G tht join two vrtis in V thn G is si to th sugrph inu y V.Apth P uv = w 0,w,,w n is squn of istint vrtis in V suh tht u = w 0,v = w n n (w i,w i ) E for vry i n. Asupth of P uv is susqun P wj w k = w j,w j+,..., w k for som 0 j < k n. Avrtxx on P uv is ll n intrnl no of P uv if x u, v. G is onnt if h pir of vrtis of G longs to pth, othrwis G is isonnt.asts of vrtis in G is ll n inpnnt st of G if th vrtis in S r pirwis non-jnt. A grph G =(V,E) is iprtit grph if V n xprss s th union of two inpnnt sts; h inpnnt st is ll prtit st. Aomplt iprtit grph is iprtit grph whr two vrtis r jnt if n only if thy r in iffrnt prtit sts. A yl of G is squn of istint vrtis strting n ning t th sm vrtx suh tht two vrtis r jnt if thy ppr onsutivly in th list. A tr T is onnt grph with no yl. Vrtis of gr on in T r ll lvs, n th rsts r ll intrnl nos. AtrT is wight if h g is ssign numr s th wight of th g. A sutr inu y st of lvs of
402 M.N. Ynhon, M.S. Byzi, n M.S. Rhmn T is th miniml sutr of T whih ontins thos lvs. Figur 2 illustrts tr T with six lvs u, v, w, x, y n z, whr th gs of th sutr of T inu y u, v n w is rwn y thik lins. W not y T uvw th sutr of tr inu y thr lvs u, v n w. On n osrv tht T uvw hs xtly on vrtx of gr 3. W ll th vrtx of gr 3 in T uvw th or of T uvw.thvrtxo is th or of T uvw in Fig. 2. Th istn twn two vrtis u n v in T, not y T (u, v), isth sum of th wights of th gs on P uv. In this ppr w hv onsir only wight trs. W us th onvntion tht if n g of tr hs no numr ssign to it thn its fult wight is on. A str is tr whr vry lf hs ommon nighor whih w ll th s of th str. w y u o x z Fig. 2. Illustrtion for lf inu sutr v AgrphG =(V,E) is ll phylognti k-powr grph if thr xists tr T suh tht h lf of T orrspons to vrtx of G nng(u, v) E if n only if T (u, v) k,whrk is givn proximity thrshol. Stinr k-powr grphs xtn th notion of phylognti k-powr. For Stinr k-powr grph th orrsponing tr my hv som intrnl nos s wll s th lvs tht orrspon to th vrtis of th grph. Both Stinr k-powr grphs n phylognti k-powr grphs long to th wily known fmily of grph powrs. Anothr spil s of grph powrs is th tr powr grph. A grph G =(V,E) is si to hv tr powr for rtin proximity thrshol k if tr T n onstrut on V suh tht (u, v) E if n only if T (u, v) k. 3 Not All Grphs Ar PCGs In this stion w show tht not ll grphs r pirwis omptiility grphs, s in th following thorm. Thorm. Not ll grphs r pirwis omptiility grphs. To prov th lim of Thorm w n th following lmms. Lmm. Lt T n g wight tr, n u, v n w thr lvs of T suh tht P uv is th lrgst pth in T uvw.ltx lf of T othr thn u, v n w. Thn, T (w, x) T (u, x) or T (w, x) T (v, x). Proof. Lt o th or of T uvw. Thn h of th pths P uv, P uw n P wv is ompos of two of th thr supths P uo, P ow n P ov.sin T (u, v) is th lrgst pth
Disovring Pirwis Comptiility Grphs 403 in T uvw, T (u, v) T (u, w). Thisimplistht T (u, o) + T (o, v) T (u, o) + T (o, w). Hn T (o, v) T (o, w). Similrly, T (u, o) T (o, w) sin T (u, v) T (w, v). SinT is tr, thr is pth from x to o. Lto x th first vrtx in V (T uvw ) V (P xo ) long th pth P xo from x. Thn lrly o x is on P uo, P vo or P wo. W first ssum tht o x is on P uo, s illustrt in Fig. 3(). Thn T (v, x) T (w, x) sin T (w, x) = T (x, o)+ T (w, o), T (v, x) = T (x, o)+ T (v, o) n T (v, o) T (w, o). W now ssum tht o x is on P vo, s illustrt in Fig. 3(). Thn T (u, x) T (w, x) sin T (w, x) = T (x, o)+ T (w, o), T (u, x) = T (x, o)+ T (o, u) n T (u, o) T (w, o). W finlly ssum tht o x is on P wo, s illustrt in Fig. 3(). Thn T (u, x) = T (u, o) + T (o, o x )+ T (o x,x) n T (w, x) = T (w, o x )+ T (o x,x). As T (w, o x ) T (w, o) n T (u, o) T (w, o), T (u, x) T (w, x). Likwis, T (v, x) T (w, x). Thus, in h s, t lst on of u n v is t istn from x tht is ithr lrgr thn or quls to th istn twn w n x. w w x o x o x u o v u o v x () () w u o x o x () Fig. 3. Diffrnt positions of x v Lmm 2. Lt G = (V,E) PCG(T, min, mx ).Lt,,, n fiv lvs of T n lt,,, n fiv vrtis of G orrsponing to th fiv lvs,,, n of T, rsptivly. Lt P th lrgst pth in th sutr of T inu y th lvs,,, n, n P th lrgst pth in T.ThnGhs no vrtx x suh tht x is jnt to, n ut not jnt to n. Proof. Assum for ontrition tht G hs vrtx x suh tht x is nighor of, n ut not of n.ltx th lvs of T orrsponing to th vrtx x of G. SinP is th lrgst pth in T mong ll th pths tht onnt pir of lvs from th st {,,,, }, mx T (x, y) mx T (x, z) y {,} z {,,} y Lmm. Sin oth n r jnt to x in G, mx T (x, y) mx. y {,} This implis tht T (x, y) mx, y {,,,, }. SinP is th lrgst pth in T, mx T (x, y) T (x, ) y Lmm. Without loss of gnrlity ssum y {,}
404 M.N. Ynhon, M.S. Byzi, n M.S. Rhmn tht T (x, ) T (x, ). Sin n x r not jnt in G n T (x, ) mx, T (x, ) < min.thn T (x, ) < min sin T (x, ) T (x, ). Sin T (x, ) < min, nnot jnt to x in G, ontrition. Using Lmm 2 w now prsnt grph whih is not PCG s in th following Lmm. Lmm 3. Lt G =(V,E) grph of 5 vrtis, n lt {V,V 2 } prtition of th st V suh tht V =5n V 2 =0. Assum tht h vrtx in V 2 hs xtly thr nighors in V n no two vrtis in V 2 hs th sm thr nighors in V. Thn G is not pirwis omptiility grph. Proof. Assum for ontrition tht G is pirwis omptiility grph, i.., G = PCG(T, min, mx ) for som T, min n mx.ltp uv th longst pth in th sutr of T inu y th lvs of T rprsnting th vrtis in V. Clrly u n v r lvs of T.Ltu n v th vrtis in V orrsponing to th lvs u n v of T, rsptivly. Lt P wx th longst pth in th sutr of T inu y th lvs of T orrsponing to th vrtis in V {u,v }. Clrly w n x r lso th lvs of T,nltw n x th vrtis in V orrsponing to w n x of T.Sin V =5, T hs lf y orrsponing to th vrtx y V suh tht y / {u,v,w,x }.Sin G is PCG of T, G nnot hv vrtx jnt to u,v n y ut not jnt to w n x y Lmm 2. Howvr, for vry omintion of thr vrtis in V, V 2 hs vrtx whih is jnt to only thos thr vrtis of th omintion. Thus thr is in vrtx in V 2 whih is jnt to u,v n y ut not to w n x. Hn G n not pirwis omptiility grph of T y Lmm 2, ontrition. Lmm 3 immitly provs Thorm. Figur 4 shows n xmpl of iprtit grph whih is not PCG. Quit intrstingly, howvr, vry omplt iprtit grph is PCG. It n shown s follows. Lt K m,n omplt iprtit grph with two prtit sts X = {x,x 2,x 3,,x m },ny = {y,y 2,y 3,,y n }. W onstrut str for h prtit st suh tht h lf orrspons to vrtx of th rsptiv prtit st. Thn w onnt th ss of th strs through n g s illustrt in Fig. 5. Finlly, w ssign on s th wight of h g. Lt T th rsulting tr. Now on n sily vrify tht K m,n = PCG(T,3, 3). 2 3 4 5 6 7 8 9 0 2 3 4 5 Fig. 4. Exmpl of grph whih is not PCG
Disovring Pirwis Comptiility Grphs 405 x y x 2 x 3 y 2 y 3 x m y n Fig. 5. Pirwis omptiility tr T of omplt iprtit grph K m,n Tking th grph sri in Lmm 3 s sugrph of lrgr grph, w n show lrgr lss of grphs whih is not PCG, s sri in th following lmm. Lmm 4. Lt G =(V,E) grph, n lt V n V 2 two isjoint susts of vrtis suh tht V =5n V 2 =0. Assum tht h vrtx in V 2 hs xtly thr nighors in V n no two vrtis in V 2 hs th sm thr nighors in V. Thn G is not pirwis omptiility grph. Proof. Assum for ontrition tht G is PCG, i.., G = PCG(T, min, mx ) for som T, min n mx.lth th sugrph of G inu y V V 2.Now,ltT H th sutr of T inu y th lvs rprsnting th vrtis in V V 2. Aoring to th finition of lf inu sutr, for ny pir of lvs u, v in T H, TH (u, v) = T (u, v). ThnH = PCG(T H, min, mx ) sin G = PCG(T, min, mx ).Howvr, H is not PCGy Lmm 3, ontrition. 4 Vrints of Tr Powr Grphs n PCGs In this stion w will show tht tr powr grphs n two of thir xtnsions r PCGs. Tr powr grphs n thir xtnsions (Stinr k-powr n phylognti k-powr grphs) hv striking rsmln, in thir unrlying onpt, with PCGs. But os this similrity signify ny rl rltionship? It os in: w fin tht tr powr grphs n ths two xtnsions r ssntilly PCGs. To stlish this rltionship of formntion thr grph lsss with pirwis omptiility grphs, w introu gnrliz grph lss whih w ll tr omptil grphs. A grph G =(V,E) is tr omptil grph if thr xists tr T suh tht ll lvs n sust of intrnl nos of T orrspon to th vrtx st V of G, n for ny two vrtis u, v V ; (u, v) E if n only if k min T (u, v) k mx.hrk min n k mx r rl numrs. W ll G th tr omptil grph of T for k min n k mx. It is quit vint from this finition tht tr omptil grph ompriss tr powr grphs, Stinr k-powr grphs, n phylognti k-powr grphs. W now hv th following thorm. Thorm 2. Evry tr omptil grph is pirwis omptiility grph. Proof. Lt G tr omptil grph of tr T for non-ngtiv rl numrs k min n k mx. Thn to prov th lim, it is suffiint to onstrut tr T n fin two non-ngtiv rl numrs min n mx suh tht G = PCG(T, min, mx ).
406 M.N. Ynhon, M.S. Byzi, n M.S. Rhmn Clrly G = PCG(T, min, mx ) for T = T, min = k min n mx = k mx if vry vrtx in V orrspons to lf in T. W thus ssum tht V ontins vrtx whih orrspons to n intrnl no of T. In this s w onstrut tr T from T s follows. For vry intrnl no u of T tht orrspons to vrtx in V, w introu surrogt intrnl no u. In ition, w trnsform u into lf no y onnting u through n g of wight λ with u. Figur 6 illustrts this trnsformtion. Hr, in ition to th lvs of T, two intrnl nos n orrspon to th vrtis in V. T is th moifi tr ftr trnsforming n into lf nos y rpling thm y n, rsptivly. f g f g h h λ λ i i () () Fig. 6. () T n () T Th formntion trnsformtion trnsmuts th sust of intrnl nos of T tht prtiipts in V into sust of lvs in T.Ltu n v two ritrry nos in T. If u n v r oth lvs in T thn T (u, v) = T (u, v). If oth u n v r intrnl nos of T tht r ontriuting to V thn in T thy r two lf nos, n T (u, v) = T (u, v)+2λ. Finlly, if only on of u n v is trnsform to lf thn T (u, v) = T (u, v) +λ. Wnxtfin min = k min n mx = k mx +2λ. Sin vry vrtx u V is rprsnt s lf in T, T my pirwis omptiility tr of G. W will prov tht T is in pirwis omptiility tr y showing tht G = PCG(T, min, mx ) for n pproprit vlu of λ. Not tht w nnot simply ssign λ =0us, in th ontxt of root fining s wll s phylogntis, n g of zro wight is not mningful. For xmpl, if n volutionry tr ontins zro wight gs thn w my fin pth of lngth zro twn two iffrnt orgnisms, whih is lrly unptl. Thrfor, w hv to hoos vlu for λ mor intlligntly. Aoring to th finition of tr omptil grphs, for vry pir of vrtis u, v V, (u, v) E if n only if k min T (u, v) k mx. Mnwhil, w hv riv T from T in suh wy tht ithr th istn twn u n v in T rmins th sm s in T, or inrs y t most 2λ. Thrfor, if w n prov tht min T (u, v) mx if n only if k min T (u, v) k mx thn it will imply tht G = PCG(T, min, mx ). Dpning on th ntur of th hng in th istn twn u n v from T to T, w hv to onsir thr iffrnt ss.
Disovring Pirwis Comptiility Grphs 407 f g f g g h h 0.4 f 0.4 i i () () () Fig. 7. () A tr omptil grph, () th orrsponing tr T, n () th orrsponing pirwis omptiility tr T Cs : T (u, v) = T (u, v). In this s, thr possil rltionships n xist mong T (u, v), k min n k mx. First, if T (u, v) <k min thn T (u, v) < min sin min = k min. Nxt, if k min T (u, v) k mx thn k min T (u, v) k mx +2λ. Tht implis, min T (u, v) mx. Finlly, lt T (u, v) >k mx. Suppos p is th minimum iffrn twn k mx n th lngth of pth in T tht is longr thn k mx,thtisp= min { T (u, v) u,v V k mx }.Thn T (u, v) k mx p. By sutrting 2λ from oth si of th inqulity w gt, T (u, v) k mx 2λ p 2λ. Whih implis T (u, v) mx p 2λ. Thrfor, if w n nsur tht p>2λthn T (u, v) will lrgr thn mx. Cs 2: T (u, v) = T (u, v)+2λ. In this s, w hv to onsir thr snrios s w hv in s. First, if k min T (u, v) k mx thn k min T (u, v)+2λ k mx +2λ. Whih implis min T (u, v)+2λ mx. Hn min T (u, v) mx. Nxt, if T (u, v) >k mx thn ing 2λ in oth sis w gt T (u, v) +2λ > k mx +2λ. Tht implis T (u, v) > mx. Finlly, lt ssum tht T (u, v) <k min. Suppos q is th minimum iffrn twn k min n th lngth of pth in T tht is smllr thn k min ; tht is q = min {k min T (u, v)}. Thnk min T (u, v) q. Sutrting 2λ u,v V from oth sis of th inqulity w gt k min T (u, v) 2λ q 2λ. Whih implis min T (u, v) q 2λ. Thrfor, if w n nsur tht q > 2λ thn T (u, v) < min. Cs 3: T (u, v) = T (u, v)+λ. This s is similr to s 2. By following th sm rsoning s in s 2, w n show tht min T (u, v) mx if n only if k min T (u, v) k mx, provi q λ. If w n stisfy th inqulity riv from s 2 (q > 2λ) thn th inqulity q>λwill immitly stisfi. From our nlysis of th thr ss ov, it is vint tht if w n stisfy th two inqulitis p>2λn q>2λsimultnously thn G = PCG(T, min, mx ).W n o this y ssigning λ ny vlu smllr thn min(p, q)/2. Thus T is pirwis omptiility tr of G, n hn G is PCG.
408 M.N. Ynhon, M.S. Byzi, n M.S. Rhmn Figur 7() illustrts n xmpl of tr omptil grph G = (V,E) n th orrsponing tr T is pit in Fig. 7(). Hr k min = 2, k mx = 4,nth wight of vry g is on. Two intrnl nos n long with th lvs of T orrspon to th vrtis in V of G. W now trnsfr T into T oring to th prour sri in Thorm 2. Figur 7() illustrts this trnsformtion. Hr, p = q =n hn w n hos ny positiv vlu lss thn 0.5 for λ. Ltλ =0.4 n thn, min = k min =2n mx = k mx +2λ =4.8. On n now sily vrify tht G = PCG(T, 2, 4.8). 5 Conlusion In this ppr, w hv provtht ll grphsr not PCGs. Aitionlly, w hv prov tht tr powr grphs n two of thir xtnsions r PCGs. Our first proof stlishs nssry onition ovr th jny rltionships tht grph must stisfy to PCG. Howvr, omplt hrtriztion of PCGs is not known. W lft it s futur work. It woul quit hllnging n signifint to vlop ffiint lgorithms for solving pirwis tr onstrution prolm for othr lsss of grphs. Suh lgorithms my om hny in oth liqu fining n volutionry rltionships moling ontxts. Aknowlgmnt This work is prt of n ongoing Mstr s thsis whih is on in Grph Drwing & Informtion Visuliztion Lortory of th Dprtmnt of CSE, BUET stlish unr th projt Fility Upgrtion for Sustinl Rsrh on Grph Drwing & Informtion Visuliztion support y th Ministry of Sin n Informtion & Communition Thnology, Govrnmnt of Bnglsh. W thnk BUET for proviing nssry supports. Rfrns. Cormn, T.H., Lisrson, C.E., Rivst, R.L., Stin, C.: Introution to Algorithms, 2n n. Th MIT Prss, Cmrig (200) 2. Jons, N.C., Pvznr, P.A.: An Introution to Bioinformtis Algorithms. Th MIT Prss, Cmrig (2004) 3. Krny, P., Munro, I., Phillips, D.: Effiint Gnrtion of Uniform Smpls from Phylognti Trs. In: Bnson, G., Pg, R.D.M. (s.) WABI 2003. LNCS (LNBI), vol. 282, pp. 77 89. Springr, Hilrg (2003) 4. Lsk, A.M.: Introution to Bioinformtis. Oxfor Univrsity Prss, Oxfor (2002) 5. Lin, G.H., Jing, T., Krny, P.E.: Phylognti k-root n stinr k-root. In: L, D.T., Tng, S.-H. (s.) ISAAC 2000. LNCS, vol. 969, pp. 539 55. Springr, Hilrg (2000) 6. Phillips, D.: Uniform Smpling From Phylogntis Trs. Mstr s thsis, Univrsity of Wtrloo (August 2002) 7. Ynhon, M.N., Hossin, K.S.M.T., Rhmn, M.S.: Pirwis omptiility grphs. Journl of Appli Mthmtis n Computing 30, 479 503 (2009)