No-bend Orthogonal Drawings of Series-Parallel Graphs

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1 No-nd Orthoonl Drwns of Srs-Prlll Grphs (Etndd Astrt) Md. Sdur Rhmn, Nortsuu E, nd Tko Nshzk Dprtmnt of Computr Sn nd Ennrn, Bnldsh Unrsty of Ennrn nd Thnoloy (BUET), Dhk 000, Bnldsh Grdut Shool of Informton Sns, Tohoku Unrsty, Ao-ym 05, Snd , Astrt. In no-nd orthoonl drwn of pln rph, h rt s drwn s pont nd h d s drwn s snl horzontl or rtl ln smnt. A plnr rph s sd to h no-nd orthoonl drwn f t lst on of ts pln mddns hs no-nd orthoonl drwn. Ery srs-prlll rph s plnr. In ths ppr w lnr-tm lorthm to mn whthr srs-prlll rph G of th mmum dr thr hs no-nd orthoonl drwn nd to fnd on f G hs. Kywords: Plnr Grph, Alorthm, Grph Drwn, Orthoonl Drwn, Bnd, SPQ tr. Introduton An orthoonl drwn of plnr rph G s drwn of G suh tht h rt s mppd to pont, h d s drwn s squn of ltrnt horzontl nd rtl ln smnts, nd ny two ds do not ross pt t thr ommon nd [NR04, RN0, RNN99, T87]. A nd s pont whr n d hns ts drton n drwn. If G hs rt of dr f or mor, thn G hs no orthoonl drwn. On th othr hnd, f G hs no rt of dr f or mor, tht s, th mmum dr of G s t most four, thn G hs n orthoonl drwn, ut my nd nds. Mnmzton of th numr of nds n n orthoonl drwn s hllnn prolm. A ndmnmum orthoonl drwn of plnr rph G hs th mnmum numr of nds mon ll possl plnr orthoonl drwns of G. Th prolm of fndn nd-mnmum orthoonl drwn s on of th most fmous prolms n th rph drwn ltrtur [BEGKLM04] nd hs n studd oth n th fd mddn sttn [RN0, RNN0, RNN99, T87] nd n th rl mddn sttn [DLV98, GT0]. Som pln rphs wth fd mddns h n orthoonl drwn wthout nds, n whh h d s drwn y P. Hly nd N.S. Nkolo (Eds.): GD 005, LNCS 84, pp , 005. Sprnr-Vrl Brln Hdlr 005

2 40 Md. S. Rhmn, N. E, nd T. Nshzk d l k j m o h n f () l k j m o h n d f () l k j m o n f d () h o n m k j l f d (d) F.. () A no-nd drwn, nd () (d) thr mddns of th sm plnr rph snl horzontl or rtl ln smnt [RNN0]. W ll suh drwn no-nd drwn of pln rph. Fur () dpts no-nd drwn of th pln rph n F. (). As rsult n th fd mddn, Rhmn t l. [RNN0] otnd nssry nd suffnt ondton for pln rph G of to h no-nd drwn, nd lnr-tm lorthm to fnd no-nd drwn f G hs. W sy tht plnr rph G hs no-nd drwn f t lst on of th pln mddn of G hs no-nd drwn. Furs (), () nd (d) dpt thr of ll pln mddns of th sm plnr rph G. Amonthmonly th mddn n F. () hs no-nd drwn s llustrtd n F. (). Thus th plnr rph G hs no-nd drwn. It s n NP-omplt prolm to mn whthr plnr rph G of 4 hs no-nd drwn n th rl mddn sttn [GT0]. Howr, for plnr rph G of, D Bttst t l. [DLV98] n O(n 5 lo n) tm lorthm to fnd ndmnmum orthoonl drwn of G. Ery srs-prlll rph s plnr rph, nd thr lorthm tks tm O(n ) for srs-prlll rph wth. Thus, y thr lorthm on n mn n tm O(n ) whthr srs-prlll rph wth hs no-nd drwn. As nothr rsult n th rl mddn, Rhmn t l. [REN05] lnr tm lorthm to mn whthr sudson G of plnr tronntd u rph hs no-nd drwn, nd to fnd no-nd drwn of G f G hs. In ths ppr w study th prolm of no-nd orthoonl drwns of srsprlll rphs wth n th rl mddn sttn, nd lnr lorthm to fnd no-nd orthoonl drwn f G hs. Th rst of th ppr s ornzd s follows. Ston dsrs som dfntons nd prsnts prlmnry rsults. Ston prsnts our lorthm to fnd no-nd drwn of onntd srs-prlll rph G f G hs. Fnlly Ston 4 s onluson. h Prlmnrs In ths ston w som dfntons nd prsnt prlmnry rsults. Lt G =(V,E) onntd rph wth rt st V nd d st E. Th dr d() ofrt s th numr of ds ndnt to n G. Wdnot th mmum dr of rph G y (G) or smply y. Th onntty κ(g) of rph G s th mnmum numr of rts whos rmol rsults n dsonntd rph or snl-rt rph K.WsythtG s k-onntd f κ(g) k.

3 No-nd Orthoonl Drwns of Srs-Prlll Grphs 4 ArphG =(V,E) s lld srs-prlll rph (wth sour s nd snk t) fthrg onsst of pr of rts onntd y snl d, or thr st two srs-prlll rphs G =(V,E ),=,, wth sour s nd snk t suh tht V = V V,E = E E, nd thr s = s, t = s nd t = t or s = s = s nd t = t = t. Apr{u, } of rts of onntd rph G s splt pr f thr st two surphs G =(V,E )ndg =(V,E ) stsfyn th follown two ondtons:. V = V V, V V = {u, }; nd.e = E E, E E =, E, E. Thus ry pr of djnt rts s splt pr. A splt omponnt of splt pr {u, } s thr n d (u, ) or mml onntd surph H of G suh tht {u, } s not splt pr of H. A splt pr {u, } of G s lld mml splt pr wth rspt to rfrn splt pr {s, t} f, for ny othr splt pr {u, }, rts s, t, u nd r n th sm splt omponnt of {u, }. Lt G onntd srs-prlll rph. Lt (s, t) ndofg. Th SPQ-tr T of G wth rspt to rfrn d =(s, t) dsrs rurs domposton of G ndud y ts splt prs [GL99]. Tr T s rootd ordrd tr whos nods r of thr typs: S, P nd Q. Ehnod of T orrsponds to surph of G, lld ts prtnnt rph G.Ehnod of T hs n ssotd onntdmultrph, lld th sklton of nd dnotd y sklton(). Tr T s rursly dfnd s follows. Trl Cs: Inthss,Gonssts of tly two prlll ds nd jonn s nd t. T onssts of snl Q-nod. Th sklton of s G tslf. Th prtnnt rph G onssts of only th d. Prlll Cs: In ths s, th splt pr {s, t} hs thr or mor splt omponnts G 0,G,,G k,k, nd G 0 onssts of only rfrn d =(s, t). Th root of T s P -nod. Thsklton() onssts of k + prlll ds 0,,, k jonn s nd t. Th prtnnt rph G = G G G k s unon of G,G,,G k.(thsklton of P -nod p n F. onssts of thr prlll ds jonn rts nd. Fur () dpts th prtnnt rph of p.) Srs Cs: In ths s th splt pr {s, t} hs tly two splt omponnts, nd on of thm onssts of th rfrn d. On my ssum tht th othr splt omponnt hs ut-rts,,, k, k, tht prtton th omponnt nto ts loks G,G,,G k n ths ordr from s to t. Thn th root of T s n S-nod. Th sklton of s yl 0,,, k whr 0 =, 0 = s, k = t, nd jons nd, k. Th prtnnt rph G of nod s unon of G,G,,G k.(thsklton of S-nod s n F. s th yl, d,,, h,,. Fur (d) dpts th prtnnt rph G s of s.) In ll ss o, w ll th d th rfrn d of nod. Ept for th trl s, nod of T hs hldrn,,, k n ths ordr; s th root of th SPQ-tr of rph G wth rspt to th rfrn d, k. W ll d th rfrn d of nod, nd ll th ndponts of d th pols of nod. Th tr otnd so fr hs Q-nod ssotd wth h

4 4 Md. S. Rhmn, N. E, nd T. Nshzk j h d s p s s 4 s (, ) p (, h) ( h, ) (, d) ( d, ) d j h f () (, j ) ( j, ) (, f ) ( f, ) (, ) s 5 (, ) (, ) f h d s p (, h) ( h, ) p (, d) ( d, ) (, j )( j, ) (, f ) ( f, ) s s 5 (, ) (, ) (, )(, ) (f) s j s 4 f h () d d d j j j f h f f h h () (d) () () (h) d F.. ()A onntd srs-prlll rph G wth =, () SPQ-tr T of G wth rspt to rfrn d (, ), nd skltons of P -nds-nods, () th prtnnt rph G s of S-nod s, (d) th prtnnt rph G s of S-nod s, () th prtnnt rph G p of P -nod p,(f)spq-trt of G wth P -nod p s th root, () th prtnnt rph of S-nod s, nd (h) th or rph of s d of G, pt th rfrn d. W omplt th SPQ-tr T y ddn Q-nod, rprsntn th rfrn d, nd mkn t th prnt of so tht t oms th root of T. An mpl of th SPQ-tr of onntd srs-prlll rph n F. () s llustrtd n F. (), whr th d drwn y thk ln n h sklton s th rfrn d of th sklton. Th SPQ-tr T dfnd o s spl s of n SPQR-tr [DT96, GL99] whr thr s no R-nodndthrootofthtrsQ-nod orrspondn to th rfrn d. On n sly modfy T to n SPQ-tr T wth n rtrry P -nod s th root s llustrtd n F. (f). In th rmndr of ths ppr, w thus onsdr SPQ-tr T wth P -nod s th root. If =, thn onntd srs-prlll rph G s yl, nd ylg hs no-nd drwn f nd only f G hs four or mor rts. On my thus ssum tht, nd tht th root P -nod of T hs thr or mor hldrn. Thn th prtnnt rph G of h nod s th surph of G ndud y th ds orrspondn to ll dsndnt Q-nod of. Th follown fts n sly drd from th ft tht h rt of G hs dr t most thr nd G hs no multpl ds.

5 No-nd Orthoonl Drwns of Srs-Prlll Grphs 4 Ft. Lt (s, t) th rfrn d of n S-nod of T,ndlt,,, k th hldrn of n ths ordr from s to t. Thn () h hld of s thr P -nod or Q-nod; () oth nd k r Q-nods; nd () nd + must Q-nods f s P -nod whr k. Ft. Eh non-root P -nod of T hs tly two hldrn, nd thr oth of th two hldrn r S-nods or on of thm s n S-nod nd th othr s Q-nod. Lt n S-nod of T,ndltund th pols of th prtnnt rph of. Lt,,, k th hldrn of n ths ordr from u to. From Ft, nd k r Q-nods. Thus nd k orrspond to ds (u, u )nd (,)ofg, rsptly. Thn th or rph for s rph otnd from th prtnnt rph of y dltn rts u nd. (Fur () llustrts prtnnt rph of S-nod s for T n F. (f), nd F. (h) llusrts or rph for s.) Vrts u nd r lld th pols of th or rph for, nd ds (u, u )nd(,) r lld hnds of th or rph for. (InFs.() nd (h) th pols of th or rph of S-nod s r rts d nd h.) For P - or Q-nod n T, w dfn th or rph for s th prtnnt rph of, nd th pols of th or rph for s th sm s th pols of th prtnnt rph of. Th or rph of P -orq-nod hs no hnd. A drwn of plnr rph G s lld n orthoonl drwn of G f h rt s mppd to pont, h d s drwn s squn of ltrnt horzontl nd rtl ln smnts, nd ny two ds do not ross pt t thr ommon nd. W ll n orthoonl drwn D of G no-nd drwn f D hs no nd, tht s, h d s drwn s snl horzontl or rtl ln smnt. A polr drwn of srs-prlll rph G s no-nd drwn of G n whh th two pols u nd of G r drwn on th outr f F o of th drwn. W ll polr drwn D of srs-prlll rph G donl drwn f D ntrsts nthr th frst qudrnt wth th orn t pol u nor th thrd qudrnt wth th orn t pol ftr rottn th drwn nd rnmn th pols f nssry, s llustrtd n F. (). Throuhout th ppr qudrnt s onsdrd to losd pln ron. Both drwn of snl rt s pont nd drwn of snl d s strht ln-smnt r donl drwns. u u u u () () () (d) F.. Polr drwns of rph G wth pols u nd : () donl drwn, () sd-on drwn, () n L-shp drwn, (d) nothr polr drwn

6 44 Md. S. Rhmn, N. E, nd T. Nshzk W ll polr drwn D of G sd-on drwn f D ntrsts nthr th frst qudrnt wth th orn t u nor th fourth qudrnt wth th orn t ftr rottn th drwn nd rnmn th pols f nssry, s llustrtd n F. (). A drwn of snl rt s pont s rrdd not to sd-on drwn, whl drwn of snl d s strht ln-smnt s sd-on drwn. A polr drwn D s lld n L-shp drwn f D ntrsts nthr th frst qudrnt wth th orn t u nor th frst qudrnt wth th orn t ftr rottn th drwn nd rnmn th pols f nssry, s llustrtd n F. (). A drwn of snl rt s pont s rrdd not to n L-shp drwn. A drwn of snl d s strht ln-smnt s not n L-shp drwn. W sy tht polr drwn s ood f t s donl, sd-on or L-shp drwn. Not ry polr drwn D s ood. For mpl, th polr drwn n F. (d) s not ood, us t s not donl, sd-on drwn or L-shp drwn. In th nt ston w n lorthm for onstrutn no-nd drwn of onntd srs-prlll rph G wth =. Ourdssfollows.LtT n SPQ-tr of G. Th or rph of h lf-nod of T onssts of snl d. For h lf-nod of T w frst drw th or rph y ln smnt s donl or sd-on drwn. Thn, n ottom up fshon, w fnd donl drwn, sd-on drwn, nd n L- shp drwn of th or rph for h ntrnl nod of T y mrn th drwns orrspondn to th hldrn of f thy st. Th drwn of th rph orrspondn to th root-nod of T ylds no-nd drwn of G f G hs polr drwn wth th splt pr, orrspondn to th root P -nod, s th pols. Our lorthm ntully hooss n pproprt SPQ-tr T of G suh tht th drwn of pln rph orrspondn to th root-nod of T ylds no-nd drwn of G f G hs. (S F. 8 for llustrton.) As w s ltr, w onstrut no-nd drwn of th or rph for nod n T y mrn th no-nd drwns of th or rphs for th hldrns of ; th no-nd drwn of th or rph for h hldrn of must polr drwn wth th two pols of th or rph. A sd-on drwn s found mor sutl for mrn thn donl drwn, nd n L-shp drwn s found mor sutl for mrn thn sd-on drwn. Intutly, to onnt th two pols y squn of horzontl nd rtl ln smnts, t lst thr turns r rqurd for donl drwn, t lst two turns r rqurd for sd-on drwn nd only on turn s rqurd for n L-shp drwn. A rph my h donl drwn lthouh t hs no sd-on or L-shp drwn nd rph my h sd-on drwn lthouh t hs no L-shp drwn. W ll polr drwn D of or rph H() fornod n T dsrl drwn f on of th follown (), () nd () holds: () D s n L-shp drwn; () D s sd-on drwn, nd H() hs no L-shp drwn; () D s donl drwn, nd H() hs nthr n L-shp drwn nor sd-on drwn. Throuhout th ppr w dnot y D() dsrl drwn of th or rph H() for nod n T.

7 No-nd Orthoonl Drwns of Srs-Prlll Grphs 45 No-nd Drwns of Bonntd Srs-Prlll Grphs In ths ston w n lorthm to onstrut no-nd orthoonl drwn of onntd srs-prlll rph G whnr G hs. If G s yl, thn t s sy to fnd no-nd drwn of G; G hs no-nd drwn f nd only f G hs four or mor rts. W thus ssum tht G s not yl. Lt T n SPQ-tr of G whos root s P -nod p hn thr hldrn. (S F. (f).) W now h th follown lmm. Lmm. Lt G srs-prlll rph wth, ltt n SPQ-tr wth P -nod p s th root, nd lt non-root nod n T.Ifthor rph H() of hs no-nd drwn, thn th follown () nd () hold: () H() hs sd-on or donl drwn, nd hn H() hs dsrl drwn D(); nd() f dsrl drwn of H() s donl drwn, thn ry no-nd drwn of H() s donl drwn for th pols of H(). Proof. W wll pro th lm y nduton sd on T. W frst ssum tht s lf-nod, tht s, Q-nod. In ths s H() onssts of snl d =(u, ), nd u nd r th pols of H(). W thus drw s snl rtl ln smnt, whh s sd-on drwn D() ofh(). Sn H() hs no L-shp drwn, D() s dsrl drwn. Thus () nd () hold. W nt ssum tht s n nnr nod othr thn th root p nd tht H() hs no-nd drwn. Lt u nd r th pols of H(). Lt,,, k (k ) th hldrn of n ths ordr from u to. SnH() hs no-nd drwn, h H( ) hs no-nd drwn. Thus w suppos ndutly tht () nd () hold for h hld of. W now h two ss to onsdr. Cs : s n S-nod. Suppos tht hs tly two hldrn. Thn H() onssts of snl rt. W drw H() s pont. Thn th donldrwns dsrl drwn D(). Thus () nd () hold. W thus ssum tht hs tly k hldrn nd k. Thn H() = H( ) H( ) H( k ), whr H( ) s th or rph of.thhypothss mpls tht, for h, k, () nd () hold for th or rph H( ). W now h th follown four suss to onsdr. Cs (): k =. In ths s H() =H( ), hn () nd () hold for H(). Cs (): k =4. In ths s H() =H( ) H( ). Ft () mpls tht thr oth nd r Q-nods or on of thm s P -nod nd th othr on s Q-nod. If nd r Q-nods, thn w n onstrut oth n L-shp drwn nd sd-on drwn of H(), s llustrtd n Fs. 4() nd 5(). Thus dsrl drwn of H() s n L-shp drwn, nd hn () nd () hold. W thus ssum tht on of thm, sy,sp -nod nd th othr s Q-nod.

8 46 Md. S. Rhmn, N. E, nd T. Nshzk () () () (d) F. 4. Dsrl drwns of th or rph for S-nods wth four hldrn W frst onsdr th s whr dsrl drwn D( )ofh( )s donl drwn. In ths s w n onstrut sd-on drwn D() ofh() s llustrtd n F. 4(). Sn th dsrl drwn of H( ) s donl drwn, H( ) hs nthr n L-shp drwn nor sd-on drwn, nd hn lrly H() hs no L-shp drwn. Thrfor th sd-on drwn D() of H() s dsrl drwn. Hn () nd () hold. W nt onsdr th s whr th dsrl drwn D( )ofh( )s sd-on drwn. Thn w n onstrut oth n L-shp drwn D() nd sd-on drwn of H() s llustrtd n Fs. 4() nd 5(). Hn () nd () hold. W fnlly onsdr th s whr th dsrl drwn D( )ofh( )s n L-shp drwn. Thn w n onstrut n L-shp drwn D() of H() s llustrtd n F. 4(d). H( ) hs sd-on or donl drwn. From t on n sly onstrut sd-on drwn of H() s llustrtd n Fs. 5() nd (). Thrfor () nd () hold. () () () F. 5. Sd-on drwns of th or rph for S-nods wth four hldrn Cs (): k =5. In ths s, H = H( ) H( ) H( 4 ). Ft () mpls tht t lst on of, nd 4 s Q-nod. In ths s w n onstrut no-nd drwn of H() suh tht () nd () hold. Th dtls r omttd n ths tndd strt. Cs (d): k 6. In ths s H = H( ) H( ) H( k ),k 6. Ft () mpls tht thr r two or mor Q-nods mon,, k. Thrfor w n sly

9 No-nd Orthoonl Drwns of Srs-Prlll Grphs 47 onstrut oth n L-shp drwn nd sd-on drwn D of H(), nd hn () nd () hold. Cs : s P -nod. In ths s k = nd hs tly two hldrn nd. Thn th hypothss mpls tht, for =,, () nd () hold for H( ). By Ft thr oth nd r S-nods or on of nd s n S-nod nd th othr s Q-nod. W frst ssum tht on of nd,sy,sq-nod, thn w h th follown two suss. Cs (): Th dsrl drwn D( ) of H( ) s donl drwn. In ths s H( ) hs nthr n L-shp drwn nor sd-on drwn. Furthrmor, ry no-nd drwn of H( ) s donl drwn y nduton hypothss. Thn D( ),D( ) nd th drwns of hnds of H( ) nnot mrd wthout nds s llustrtd n F. 6(). Thrfor H() dosnoth no-nd drwn, ontrry to th ssumpton tht H() hs no-nd drwn. Thrfor ths s dos not our. Cs (): Th dsrl drwn D( ) of H( ) s sd-on or L-shp drwn. In ths s w n onstrut no-nd drwn D() ofh() suh tht () nd () hold s llustrtd n Fs. 6() (). Q.E.D. W ll th lorthm dsrd n th proof of Lmm for fndn dsrl drwn D() ofh() Alorthm Dsrl-Drwn whnr H() hs nond drwn. Clrly Alorthm Dsrl-Drwn tks lnr-tm. In th rst of th ston w Alorthm Bonntd-Drw for fndn no-nd drwn of G whnr G hs. Rmmr tht th root nod p n T donl drwn no no nd drwn () donl drwn (d) sd on drwn sd on drwn () sd on drwn () sd on drwn () L shp drwn sd on drwn () L shp drwn (f) L shp drwn (h) L shp drwn () D( ) ln donl drwn sd on drwn L shp drwn D( ) Q nod S nod F. 6. Drwns of H() forp -nod p

10 48 Md. S. Rhmn, N. E, nd T. Nshzk hs thr hldrn s dptd n F. (f). Lt, nd th thr hldrn of p n T.IfGhs no-nd drwn, thn H( ),, hs no-nd drwn. For, w fnd dsrl drwn D( )ofh( ) y Alorthm Dsrl-Drwn. IfG hs polr drwn for th pols orrspondn to p, thn w now fnd no-nd drwn of G = H( p ) y mrn th drwns of D( ),D( ),D( ) nd th drwns of thr hnds. Othrws, w fnd pproprt pols for whh G hs no-nd polr drwn. Sn G s smpl rph, t most on of, nd s Q-nod. W now h th follown two ss to onsdr. Cs : on of thm, sy,sq-nod. In ths s only s Q-nod. If t lst on of D( )ndd( )s donl drwn, Thn G dos not h no-nd drwn s llustrtd n F. 7()-(). Othrws, G hs no-nd drwn s llustrtd n F. 7(d)-(f). Th dtls r omttd. donl drwn no no nd drwn () sd on drwn L shp drwn no no nd drwn () no no nd drwn () (d) () (f) D( ) D( ) donl drwn sd on drwn L shp drwn F. 7. Illustrton for Cs of Alorthm Bonntd-Drw Cs : ll of, nd r S-nods. If t most on of D( ),D( )ndd( ) s donl drwn, thn w n sly onstrut no-nd drwn of G.IfllofD( ),D( )ndd( ) r donl drwns, thn on n sly osr tht G dos not h no-nd drwn. W thus onsdr th s whr tly two of D( ),D( )ndd( )r donl drwns. If two of D( ),D( )ndd( ) r donl drwns nd th othr s n L-shp drwn, thn lrly w n onstrut no-nd drwn of G. W my thus ssum tht two of D( ),D( )ndd( ) r donl drwns nd th othr s sd-on drwn. W my ssum wthout loss of nrlty tht D( )ndd( ) r donl drwns nd D( ) s sd-on drwn. By Lmm () ry no-nd drwn of h of H( )ndh( ) s donl drwn. By mrn D( )ndd( )

11 No-nd Orthoonl Drwns of Srs-Prlll Grphs 49 p P S S S donl donl sd on P 4 5 sd on 6 S 4 5 sd on S sd on D( ) D( ) D( ) () () () (d) P S S S S donl 4 5 donl sd on sd on p P donl S donl D ( ) D( ) D( ) () (f) " () (h) F. 8. () (d) A no-nd drwn of G nnot found usn tr T, nd () (h) no-nd drwn of G n found usn tr T w n otn only donl drwn D.SnD( ) s sd-on drwn, D nd D( ) nnot mrd to produ no-nd drwn of G. Howr, w n onstrut no-nd drwn of G f H( ) hs nothr pproprt no-nd drwn. W n llustrt mpl n Fur 8 nd omt th dtls of th proof. G hs no polr drwn wth th pols orrspondn to p s llustrtd n F. 8(d). Howr, G my h no-nd drwn whn on onsdrs som othr splt pr s pols. W thrfor onsdr n SPQ-tr T of G wth s th root, s llustrtd n F. 8(f), whr, 4 nd 5 r th hldrn of.ehof D( 4 )ndd( 5 ) rmns sm s on otnd for th SPQ-tr T. Consdrn T, D( ) s donl drwn D. W n thus fnd no-nd drwn of G y rursly pplyn Alorthm Bonntd-Drw rrdn D( ),D( 4 ) nd D( 5 )snwd( ),D( )ndd( ), rsptly. (Fur 8(h) shows tht G hs no-nd polr drwn wth th pols orrspondn to root.) If w nnot drw no-nd orthoonl drwn of G y rptn th oprton o, thn G dos not h no-nd drwn. Thus Alorthm Bonntd-Drw fnds no-nd drwn of G f G hs. On n ffntly mplmnt Alorthm Bonntd-Drw so tht t tks tm O(n). Th dtls r omttd n ths tndd strt. Thorm. Lt G onntd srs-prlll rph of th mmum dr thr. Thn Alorthm Bonntd-Drw fnds no-nd drwn of G n tm O(n) whnr G hs, whr n s th numr of rts of G.

12 40 Md. S. Rhmn, N. E, nd T. Nshzk 4 Conlusons In ths ppr, w lnr-tm lorthm to fnd no-nd drwn of onntd srs-prlll rph G of mmum dr t most thr. W lso n lorthm to fnd no-nd drwn of srs-prlll rph G whh s not lwys onntd. Howr, th lorthm s omttd n ths tndd strt du to p lmtton. It s lft s futur work to fnd nd-mnmum drwn of srs-prlll rphs nd to fnd lnr-tm lorthm for lrr lss of plnr rphs. Rfrns [BEGKLM04] F. Brndnur, D. Eppstn, M. T. Goodrh, S. Koouro, G. Lott nd P. Mutzl, Sltd opn prolms n rph drwns, Pro. ofgd 0, Lt, Nots n Computr Sn, 9, pp , 004. [DLV98] G. D Bttst, G. Lott nd F. Vru, Sprlty nd optml orthoonl drwns, SIAM J. Comput., 7(6), pp , 998. [DT96] G. D Bttst nd R. Tmss, On-ln plnrty tstn, SIAMJ. Comput., 5(5), pp , 996. [GL99] A. Gr nd G. Lott, Almost nd-optml plnr orthoonl drwns of onntd dr- plnr rphs n qudrt tm, Pro.of GD 99, Lt. Nots n Computr Sn, 7, pp. 8-48, 999. [GT0] A. Gr nd R. Tmss, On th omputtonl omplty of upwrd nd rtlnr plnrty tstn, SIAM J. Comput., (), pp , 00. [NR04] T. Nshzk nd M. S. Rhmn, Plnr Grph Drwn, World Sntf, Snpor, 004. [REN05] M. S. Rhmn, N. E nd T. Nshzk, No-nd orthoonl drwns of sudsons of plnr tronntd u rphs, IEICE Trns. Inf. & Syst., E88-D (), pp.-0, 005. [RN0] M. S. Rhmn nd T. Nshzk, Bnd-mnmum orthoonl drwns of pln -rphs, Pro. of WG 0, Lt. Nots n Computr Sn, [RNN0] 57, pp , 00. M. S. Rhmn, M. Nznn nd T. Nshzk, Orthoonl drwns of pln rphs wthout nds, Journl of Grph Al. nd Appl., 7(4), pp. 5-6, 00. [RNN99] M.S. Rhmn, S. Nkno nd T. Nshzk, A lnr lorthm for nd-optml orthoonl drwns of tronntd u pln rphs, Journl of Grph Al. nd Appl., (4), pp. -6, 999. [T87] R. Tmss, On mddn rph n th rd wth th mnmum numr of nds, SIAM J. Comput., 6, pp , 987.

The University of Sydney MATH 2009

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