Witness-Bar Visibility Graphs

Transcription

1 Mxin Confrn on Disrt Mthmtis n Computtionl Gomtry Witnss-Br Visiility Grphs Crmn Cortés Frrn Hurto Alrto Márquz Jsús Vlnzul Astrt Br visiility grphs wr introu in th svntis s mol for som VLSI lyout prolms. Thy hv n lso stui sin thn y th grph rwing ommunity, n rntly svrl gnrliztions n rstrit vrsions hv n propos. W introu gnrliztion, witnss-r visiility grphs, n w prov tht this lss nompsss ll th r-visiility vritions onsir so fr. In ition, w show tht mny lsss of grphs r ontin in this fmily, inluing in prtiulr ll plnr grphs, intrvl grphs, irulr r grphs n prmuttion grphs. 1 Introution n prliminry finitions Givn st S of isjoint horizontl lin sgmnts in th pln (ll rs hrftr) w sy tht G is r-visiility grph if thr is ijtion twn S n th vrtis of G, n n g twn two of ths if n only if thr is vrtil sgmnt (ll lin of sight) twn th orrsponing rs tht os not intrst ny othr r. W lso sy tht S is r visiility rprsnttion (or r visiility rwing) of G. Br visiility grphs wr introu y Gry, Johnson n So [15] s moling tool for igitl iruit sign (s lso [20]). Ths rprsnttions r lso usful tool for isplying igrms tht onvy visul informtion on rltions mong t, whih is why mny vritions of ths grphs hv n onsir y th grph rwing ommunity [7, 9, 10, 11, 14, 17, 18]. W n som finitions for w n pos prisly our prolm; w us stnr trminology s in [6]. W ll v-sgmnt ny vrtil sgmnt. W ll ε-sgmnt ny xis lign rtngl hving with ε > 0 (intuitivly, thik vrtil sgmnt). Lt s n t two horizontl rs. W sy tht v-sgmnt onnts s n t if its npoints r in s n t. W sy tht n ε-sgmnt onnts s n t if its horizontl sis r ontin in s n t. Lt S st of non-ovrlpping horizontl sgmnts (rs). Two rs s, t S r visil if, n only if, thr is v-sgmnt onnting s n t intrsting no othr sgmnt in S. W sy tht s n t r ε-visil if, n only if thr is n ε sgmnt onnting s n t intrsting no othr sgmnt in S. With th pring finition, r visiility grphs s fin in th first prgrph of this stion tk s nos st of isjoint rs, n thr is n g twn two nos if n only if th orrsponing rs r visil (this is lso ll strong visiility rprsnttion of th grph [21]). If inst of visiility w rquir ε-visiility, thn w gt r ε-visiility grphs or, quivlntly, n ε-visiility rprsnttion of th grph. Th lttr hv n hrtriz s thos grphs tht mit plnr ming with ll utpoints on th xtrior f [21, 22]. A grph G is wk r visiility grph if its nos n put in ijtion with st of isjoint rs n th nos orrsponing to vry g in G r ε-visil (not tht not vry ε-visiility n n g). This fmily of grphs is xtly th lss of ll plnr grphs [12]. Finlly, w sy tht G is r k-visiility grph if thr is ijtion twn st of rs S n th vrtis of G, n n g twn two of ths if n only if thr is v-sgmnt joining th orrsponing rs tht intrsts t most k othr rs. This gnrliztion hs n introu in rnt yrs [10, 14]. Dprtmnto Mtmáti Apli I, Univrsi Svill Dprtmnt Mtmàti Apli II, Univrsitt Politèni Ctluny. Prtilly support y projts MINECO MTM , Gn. Ctluny DGR 2009SGR1040, n ESF EUROCORES progrmm EuroGIGA, CRP ComPoS: MICINN Projt EUI- EURC , for Spin.

2 Jorg Urruti s Fst, Ox, Mxio, Novmr 11-15, 2013 In this ppr w introu strongr gnrliztion, witnss-r visiility grphs, n w prov tht this rprsnttion pproh nompsss ll th r-visiility vritions onsir so fr. In ition, w show tht mny lsss of grphs r ontin in this fmily, inluing in prtiulr ll plnr grphs, intrvl grphs, irulr r grphs n prmuttion grphs. For th finition of witnss-r visiility grphs w onsir, in ition to th st S of rs tht r in orrsponn on-to-on with th vrtis of th grph ing onstrut, st of grn rs tht fvor visiility, n st of r rs tht ostrut visiility. Grn rs t s positiv witnsss whil r rs orrspon to ngtiv witnsss. Th rs from S nithr fvor nor ostrut visiilitis. For th s of sription it is usful to onsir lso purpl rs tht ostrut visiility in slightly iffrnt wy thn r rs. Dfinition 1 Lt S, S G, S P n S R four sts of horizontl sgmnts (rs, grn-rs, purpl-rs, n r-rs, rsptivly) suh tht ny two lmnts in S S G S P S R r isjoint. W fin: 1. Th grn-r visiility grph of S with rspt to S G hs on vrtx for vry lmnt in S, n two rs s, t S r jnt if n only if thr is n ε-sgmnt onnting s n t tht rosss t lst on grn r. 2. Th purpl-r visiility grph of S with rspt to S P hs on vrtx for vry lmnt in S, n two rs s, t S r jnt if n only if thr is n ε-sgmnt onnting s n t tht os not ross ny purpl r. 3. Th witnss-r visiility grph of S with rspt to S G n S R hs on vrtx for vry lmnt in S, n two rs s, t S r jnt if n only if thr is n ε-sgmnt onnting s n t tht rosss stritly mor grn rs thn r rs. Th lss of grn, purpl n witnss-r visiility grphs r not, rsptivly, y GBG, PBG n WBG. An illustrtion of th thr typs of grphs is shown in Figur 1 (on lk n whit printr, nors ppr s thin lins, r rs s thik rk lins, purpl lins s thik lins olor light gry, n th grn lins r sn s thik strip lins). x z y x z y x z y Figur 1: Exmpls of grphs in th fmilis GBG, PBG n WBG, for th st of rs S = {,,,, }. This work is vot to th stuy of th lsss of grphs tht n rprsnt vi grn, purpl or r-visiility grphs n its proprtis. W strt y onsiring th lsss GBG n PBG, whih will prov to sulsss of WBG. Thn w will numrt lsss of grphs tht r ontin in WBG, s wll s proprtis of this lss rlt to plnrity. Th trminology witnss-r visiility grphs is inspir y th onpt of witnss proximity grphs, whih fouss on iing nighorlinss rltions mong points in finit st oring to th prsn of som positiv n/or ngtiv witnss points, topi tht hs n stui in rnt yrs [1, 2, 3, 4, 13].

3 Mxin Confrn on Disrt Mthmtis n Computtionl Gomtry 2 PBG n GBG r sulsss of WBG W prov in this stion tht th lsss PBG n GBG tht w hv introu for th sk of lrity in mny proofs, r sulsss of WBG, th lss tht is our tul sujt of stuy. It is worth notiing first tht if w onsir grph in WBG tht hs only grn rs, thn it is xtly grph in GBG. Howvr, if w hv grph in WBG tht hs only r rs, thn it is just n mpty grph, us for visiility w rquir th grn rs ross y lin of sight to stritly mor thn th ross r rs. This is why w lso onsir th lss PBG in whih purpl rs ply strongr ostrution rol. Lmm 1 Th lss of grphs WBG ontins th lsss GBG n PBG. Proof. (Skth) Th ft tht GBG WBG is trivil; th grn-r visiility rprsnttion is lso witnss-r visiility rprsnttion. In Figur 2 w illustrt mtho to otin WBG rprsnttion of PBG grph. u 1 u 1 u 1 u 5 u 2 u 2 u 3 u 4 u 2 u 3 u 4 u 3 u4 ) ) ) u 5 u 5 Figur 2: ) n ): A grph n on of its purpl-r visiility rprsnttions. Visiility orriors (ε-sgmnts) twn jnt vrtis r shown. ) Th grn-r visiility rprsnttion rri out from ). Th smll r r twn u 4 n u 5 is to voi th jny twn u 1 n u 5 us y th prviously grn rs. In Stion 5 w will prov tht th two inlusions in Lmm 1 r strit. 3 Th grph lss GBG In this stion w stuy th lss GBG hving s min ojtiv to otin proprtis of th suprlss WBG. In our wy, w lso xplor som rltionships with othr grph lsss. Rll now tht n intrvl grph is th intrstion grph of st of (los) intrvls on th rl lin; this is, it hs on vrtx for h intrvl in th st, n n g twn vry pir of vrtis orrsponing to intrvls tht hv nonmpty intrstion. Thorm 2 Lt G grph. If G is n intrvl grph, thn G GBG. Th rvrs is in gnrl fls. Proof. Consir th intrvls on th rl lin tht fin G. First of ll, noti tht w n xtn infinitsimlly th intrvls in suh wy tht vry pir tht intrst ovrlp in n intrvl of positiv lngth. Thn ssign ritrry to ths intrvls iffrnt hights so tht w hv st of rs in th pln. To otin grn-r visiility rprsnttion of G it suffis to shil h sgmnt with two grn rs of th sm lngth, on ov n on low. Noti tht nothr rprsnttion using lss grn rs my lso xist. Th grph lss inlusion is strit, s on n show tht C 4 is in GBG (Figur 3) whil th intrvl grph os not hv yls of lngth grtr thn thr. A iffrn twn th lss of intrvl grphs n GBG is stt in nxt proposition, whos proof is omitt hr. Proposition 3 Th girth of ny grph in GBG is t most four, n this vlu is hivl.

4 Jorg Urruti s Fst, Ox, Mxio, Novmr 11-15, 2013 x y ) ) Figur 3: ) C 4 is GBG grph ut ) it is not n intrvl grph: n must not ovrlp. Sin is jnt to oth vrtis it ovrs th gp twn thm, s shoul o. Thrfor n must jnt in th intrvl grph. As onsqun of th prvious rsult, it follows tht th grn-r visiility grph lss os not ontin ny of th r-visiilitis lsss sri in th introution of this ppr, us C n n rprsnt s wk/ε/strong r visiilty grph for vry n 3 [21]. Lt us rll nxt som finitions [8].Th imnsion of prtilly orr st P (post) is th smllst possil numr of totl orrs whos intrstion is th prtil orr in P. Th omprility grph inu y post P = (X, ) is th grph with vrtx st X in whih x, y X r jnt if n only if ithr x y or y x in P; in othr wors, it is th unirt grph unrlying P. Altrntivly [16], omprility grph is grph suh tht vry gnrliz yl of o lngth hs tringulr hor ( gnrliz yl is los wlk tht uss h g of th grph t most on in h irtion). A omprility grph hs imnsion 2 if it is th omprility grph of post of imnsion 2 (it hs n shown tht this onpt of imnsion is wll fin [8]). Ths finitions n lso onsir from gomtri point of viw [19]. Lt R 2 th Eulin 2-imnsionl sp. A mtho of orring th points of R 2 is th following: (x i, y i ) (x j, y j ) if, n only if, x i x j n y i y j ; w sy tht (x i, y i ) is omint y (x j, y j ). A post P = (X, ) hs imnsion t most 2 if it n m in R 2 in suh wy tht th orr is prsrv. Morovr, hnging th oorints slightly if rquir, w n ssum tht th ming is suh tht no two lmnts of X hv qul x or y oorint. As onsqun, ny omprility grph of imnsion 2 n lwys thought s point st in R 2 with th ominn orr. This gomtri rprsnttion pplis in Eulin -imnsionl sp to omprility grphs of imnsion. Thorm 4 Thr r grphs in GBG tht r not omprility grphs. Proof. Th grph pit in Figur 4 ) is not omprility grph sin th gnrliz yl v 1 v 2 v 4 v 5 v 4 v 3 v 2 v 1 hs o lngth (svn) ut hs no tringulr hors. Figur 4 ) shows grn-r visiility rprsnttion of this grph. v 1 v 2 v 2 v 5 v 4 v5 v 1 v 4 v 3 v 3 ) ) Figur 4: ) A grph tht is not omprility grph n ) grn-r visiility rprsnttion of this grph.

5 Mxin Confrn on Disrt Mthmtis n Computtionl Gomtry 4 Th grph lss PBG In this stion w turn our ttntion to th lss PBG, hving gin s min ojtiv to otin proprtis of th suprlss WBG. In our wy w xplor s wll som rltionships with othr grph lsss. On my think tht th lsss GBG n PBG r rlt y omplmnttion, possily y swithing purpl n grn r oloring, ut it is not th s. For xmpl th union of two isjoint tringulr yls is in GBG, s sn in th pring stion, ut its omplmnt is K 3,3, whih is not in PBG, ft tht w prov low. Rvrsly, th ft tht G PBG os not imply tht G GBG. A simpl xmpl is otin y onsiring G = G = C 5, whih mits purpl-r visiility rprsnttion (s Figur 5), ut w hv prov in Proposition 3 tht C 5 / GBG. Figur 5: A purpl-r visiility rprsnttion of C 5. It is nithr tru tht th omplmnt of grph in PBG is in PBG. Consir G = K 3,3, whih onsists of th union of two tringls n hn G PBG (tringls r intrvl grphs) ut G = K 3,3 / PBG s will shown in Thorm 7. On th othr hn, C 5 is lso n xmpl tht th lss PBG ontins som non-prft grphs. Rll tht prft grph is grph in whih th hromti numr of vry inu sugrph quls th siz of th lrgst liqu of tht sugrph. Th lrgst liqu in C 5 is K 2, howvr χ(c 5 ) = 3, thrfor C 5 is not prft. On th positiv si, lt us s tht intrvl grphs mit purpl-r visiility rprsnttion n prov lmm tht will usful ltr. Thorm 5 If G is n intrvl grph, thn G PBG (n thrfor G WBG). Proof. Consir th intrvls on th rl lin tht fin G n lift thm to ritrry iffrnt hights so tht w hv st of rs in th pln. This is lry purpl-r visiility rprsnttion of G. Lmm 6 Lt G tringl-fr grph. If G PBG thn G is plnr grph. Proof. (Skth) Givn grph G PBG w n us th visiility winows in orr to otin pln rprsnttion of G s it is pit in Figur 6. u 4 u 4 u 2 u 5 u 7 u 2 u 5 u 7 u 6 u 6 u 3 u 8 u 3 u 8 u 1 u 1 ) ) Figur 6: A purpl-r visiility rprsnttion of tringl-fr grph n th orrsponing onstrution of its plnr ming -in lu-.

6 Jorg Urruti s Fst, Ox, Mxio, Novmr 11-15, 2013 Thorm 7 1) K 3,3 / PBG. 2) K n PBG, n. 3) Th proprty of mitting purpl-r visiility rprsnttion is not inhrit y sugrphs. Proposition 8 Thr r nonplnr grphs with tringls tht o not mit purpl-r visiility rprsnttion. An xmpl is shown in Figur 7 g h i f o p j k l m n Figur 7: A nonplnr grph G with tringl ( gjk),suh tht G / PBG. Thorm 9 Evry grph G tht n rprsnt s strong/ε/wk r visiility grph mits s wll purpl-r visiility rprsnttion (n thrfor WBG rprsnttion s wll). Proof. As vry grph tht mits th first or son rprsnttion is lso rlizl using wk visiility [21], w only hv to prov tht th lttr lss of grphs is ontin in PBG. Lt G grph rliz s wk r visiility grph. Eh no u of G is r, tht hs som visiility orriors ov n low (s Figur 8()). Osrv tht w n shrink ll th visiility orriors to still hv positiv with, yt trmining isjoint intrvls on u (Figur 8()). Aftr th shrinking, w n shil ll th rs with purpl rs on oth sis, yt without rossing th orriors of sight (Figur 8() n ()). This is lrly purpl-r visiility rprsnttion of G. ) ) ) ) Figur 8: Construting purpl-r visiility rprsnttion of r visiility grph. 5 Th lss WBG of witnss-r visiility grphs W strt this stion y osrving tht th ontntions in Lmm 1 r strit, tht is, th lsss GBG n PBG r stritly ontin in WBG. Figur 9 shows witnss-r rprsnttion of suivision of K 3,3. By Proposition 3, this grph is not in GBG sin it ontins horlss yls of lngth fiv. On th othr hn, th grph is tringl-fr n nonplnr, thrfor it is not in PBG y Lmm 6. W lim in th introution tht k-r visiility is lso gnrliz y witnss-r visiility. This is prisly wht w prov in th following thorm:

7 Mxin Confrn on Disrt Mthmtis n Computtionl Gomtry v 1 v 1 v 2 v 3 v 14 v 14 v 6 v 5 v 4 v 2 v 4 v 5 v 6 v 3 Figur 9: A witnss-r rprsnttion of suivision of K 3,3. Thorm 10 Evry grph G tht n rprsnt s r k-visiility grph mits s wll witnss-r visiility rprsnttion. Proof. Lt B th st of rs in r k-visiility rprsnttion of G; rll tht thr is n g twn two of ths if n only if thr is v-sgmnt joining th orrsponing rs tht intrsts t most k othr rs. For th sk of lrity lt us s tht, if nssry, w n moify slightly th rs in B to voi som gnris, otining gin st of rs whos orrsponing r k-visiility grph is still G. Lu us onsir vrtil lin through h npoint of ll rs in B. If for ny of ths vrtil lins l thr is st L of rs in B whos right npoint is on l, n st R of rs in B whos lft npoint is lso on l, w xtn infinitsimlly to th right ll th rs in L, without jumping ovr ny of th othr vrtil lins. Clrly th nw st of rs still inus G y r k-visiility, s nithr k-visiilitis r stroy, nor nw k-visiilitis r rt. Thrfor w n ssum, without loss of gnrlity, tht no suh ovrtilitis r prsnt in B. Lu us ll onsir th st of vrtil lins through th npoints of th rs in B. Ths lins ompos th plns into strips; insi h strip w hv stk of (portions of) rs of qul lngth, th with of th strip, pl t iffrnt hights. Lt us fous in ny of ths strips (only if it is non mpty of rs), whih w not S. Lt B S th st of rs in S n lt m th numr of rs in B S. Noti tht t this stp w isrgr rs or portions of rs outsi S. Cs 1: m k + 2. In this s w us two uxiliry vrtil lins to split S into thr vrtil strips of qul with, whih w ll sls. In th ntrl sl, twn ny two onsutiv rs, w pl grn r. Th grph WBG inu insi S is th omplt grph, whih ws xtly th sitution for r k-visiility. Cs 2: m > k + 2. Lt us ll 1,..., m th rs in B S, in orr on inrsing hight. In this s thr r m k 1 susts of k + 2 onsutiv rs in B S, nmly B 1 = { 1,..., k+2 }, B 2 = { 2,..., k+3 },..., B m k 1 = { m k 1,..., m }. Lt us suivi th strip S into 2(m k 1) + 1 = 2m 2k 1 vrtil sls, tht w not S 1,..., S 2m 2k 1. W r ling insi sl S 2i with th WBG-visiility mong th rs in B i, y pling grn r with th sm with thn S 2i twn ny two onsutiv portions of r in B i, n on stk of k + 1 r rs of th sm with just low i (ut ov i 1 ) n nothr stk of k + 1 r rs of th sm with just ov i+k+1 (ut low i+k+2 ). Th grph WBG inu insi S 2i for th rs B i is th omplt grph, n no othr WBG-visiilitis ppr in S 2i. Rpting th onstrution, w hv mimik xtly th sitution for r k-visiility using WBG-visiility, s lim. W prov nxt tht nothr intrsting lss of grphs is ontin in WBG. A irulr-r grph is th intrstion grph of st of (usully ssum to opn) rs on th irl. It hs on vrtx for h r in th st, n n g twn vry pir of vrtis orrsponing to rs tht intrst. Thorm 11 If G is irulr-r grph, thn G WBG.

8 Jorg Urruti s Fst, Ox, Mxio, Novmr 11-15, A B ) ) ) Figur 10: A witnss-r rprsnttion of irulr-r grph. Proof. (Skth) Th rliztion stps r shmtiz in th xmpl in Figur 10. W prov nxt nothr intrsting grph lss ontinmnt in WBG. Lmm 12 Lt G grph. If G is omprility grph of imnsion 2, thn G WBG. Proof. Lt X = {(x 1, y 1 ),..., (x n, y n )} th vrtx st of th ominn rliztion of G in th pln, n ssum, without loss of gnrlity, tht th vrtis hv n ll in suh wy tht x 1 < x 2 <... < x n. For i = 1,..., n, lt us now rprsnt vrtx (x i, y i ) s r in R 2 givn y th sgmnt s i with npoints (x 1, y i ) n (x n, y i ). W pl ov s i i = 1,..., n, r r ginning t x-oorint x 1 n ning t x i, n grn r ginning t x-oorint x i n ning t x n. Blow s i, w rw grn r ginning t x-oorint x 1 n ning t x i n r on ginning t x-oorint x i n ning t x n (Figur 11). It is not iffiult to s tht this is witnss-r visiility rprsnttion of G. y j y i y i y j x 1 x 2... x i... x j... x n ) x 1 x 2... x i... x j... x n ) Figur 11: Proof of Lmm 12. A prmuttion grph is th intrstion grph of fmily of lin sgmnts tht onnt two prlll lins. Equivlntly, givn prmuttion (σ 1, σ 2,..., σ n ) of th numrs 1, 2, 3,...n, prmuttion grph hs vrtx for h numr 1, 2, 3,...n n n g twn ny two numrs tht r in rvrs orr in th prmuttion, i.., n g twn ny two numrs whr th sgmnts ross in th prmuttion igrm. Th lss of prmuttion grphs hs n wily stui [8], n sin thy r hrtriz s omprility grphs unrlying prtilly orr sts tht hv imnsion t most two, w n sily infr tht thy r lso ontin in WBG.

9 Mxin Confrn on Disrt Mthmtis n Computtionl Gomtry Thorm 13 If G is prmuttion grph, thn G WBG. Givn grph G, lt G th grph rsulting from suiviing on ll gs in G. Thn w hv: Lmm 14 Lt G grph. If G WBG thn G is plnr grph. Lmm 15 K 3,3 / WBG n K 3,3 WBG. Thorm 16 Th lss of th grphs tht mit witnss-r visiility rprsnttion is not los unr omplmnttion. W onlu this stion with nothr rsult on th lss WBG, tht isrs th possiility of hrtrizing th lss y forin minors: Thorm 17 Th proprty of mitting witnss-r visiility rprsnttion is not inhrit y sugrphs. Proof. W know tht K 6 WBG from Thorm 7 n Lmm 1. On th othr hn K 3,3 is suivision of sugrph of K 6, ut w know from Lmm 15 tht K 3,3 is not in WBG. This sttls th lim. 6 Conluing rmrks Lt us summriz th proprtis w hv prov for th lss WBG of witnss-r visiility grphs: Evry grph G tht n rprsnt s strong/ε/wk r visiility grph mits s wll witnss-r visiility rprsnttion Evry grph G tht n rprsnt s r k-visiility grph mits s wll witnss-r visiility rprsnttion Th lss of intrvl grphs is ontin in th lss WBG. If G is irulr r grph, thn G WBG If G is prmuttion grph, thn G WBG Th lss of th grphs tht mit witnss-r visiility rprsnttion is not los unr omplmnttion. Th proprty of mitting witnss-r visiility rprsnttion is not inhrit y sugrphs, whih isrs th possiility of hrtrizing th grph lss WBG y forin minors. W onlu tht th grph lss WBG is vry rih n nompsss mny othr lsss. Howvr, to otin hrtriztion or rognition lgorithm ppr to quit hllnging prolms. Rfrns [1] O. Aihholzr, R. Fil, T. Hkl, A. Pilz, P. Rmos, M. vn Krvl, n B. Vogtnhur. Bloking Dluny tringultions. To ppr in Computtionl Gomtry: Thory n Applitions (pt 2012). Onlin vrsion vill t [2] B. Aronov, M. Duliu n F. Hurto, Witnss (Dluny) grphs. Computtionl Gomtry: Thory n Applitions 44(6-7): , [3] B. Aronov, M. Duliu n F. Hurto, Witnss Gril grphs. To ppr in Computtionl Gomtry: Thory n Applitions (pt 2011). Onlin vrsion t DOI: /j.omgo

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