Ostrutions to horl irulr-r grphs of smll inepenene numer Mthew Frnis,1 Pvol Hell,2 Jurj Stho,3 Institute of Mth. Sienes, IV Cross Ro, Trmni, Chenni 600 113, Ini Shool of Comp. Siene, Simon Frser University, Burny, Cn V5A 1S6 DIMAP n Mth. Institute, University of Wrwik, Coventry CV4 7AL, UK Astrt A loking quruple (BQ) is quruple of verties of grph suh tht ny two verties of the quruple either miss (hve no neighours on) some pth onneting the remining two verties of the quruple, or re onnete y some pth misse y the remining two verties. This is kin to the notion of steroil triple use in the lssil hrteriztion of intervl grphs y Lekkerkerker n Boln [10]. In this note, we first oserve tht loking quruples re ostrutions for irulrr grphs. We then fous on horl grphs, n stuy the reltionship etween the struture of horl grphs n the presene/sene of loking quruples. Our ontriution is two-fol. Firstly, we provie forien inue sugrph hrteriztion of horl grphs without loking quruples. In prtiulr, we oserve tht ll the forien sugrphs re vrints of the sugrphs forien for intervl grphs [10]. Seonly, we show tht the sene of loking quruples is suffiient to gurntee tht horl grph with no inepenent set of size five is irulr-r grph. In our proof we use novel geometri pproh, onstruting irulr-r representtion y trversing roun refully hosen lique tree. Keywors: irulr-r grph, horl grph, steroil triple, lique tree, ostrution, loking quruple, forien sugrph hrteriztion
1 Introution The stuy of grph ostrutions hs long trition in grph theory. To unerstn the struture of grphs in prtiulr grph lss, it is often useful (if not esier) inste to hrterize ll miniml grphs tht re not in the lss, usully known s ostrutions. They often result in elegnt hrteriztion theorems n n e use s suint ertifites in ertifying lgorithms. In this pper, we seek ostrutions to irulr-r grphs, the intersetion grphs of fmilies of rs of irle. This prolem tes k t lest s fr s the 1970 s [9,12,13], n remins hllenging question pturing the interest of mny reserhers over the yers [1,2,6,9,11,12,13]. Preting the stuy of irulr r grphs, the lss of intervl grphs, intersetion grphs of fmilies of intervls of the rel line, ws investigte. Intervl grphs re sulss of horl grphs, grphs in whih every yle hs hor, s well s of irulr-r grphs. They re known to mit numer of interesting hrteriztions [8,10] n effiient reognition lgorithms [4,5]. In prtiulr, the result of Lekkerkerker n Boln [10] esries intervl grphs in terms of forien inue sugrphs s well s forien sustrutures horless yles n so-lle steroil triples. This result is the min motivtion of our pper wherein we seek to esrie nlogous forien sustrutures for irulr-r grphs. We remrk in pssing tht, esies intervl grphs, there re other suses of irulr-r grphs tht hve lrey een hrterize y the sene of simple ostrutions. Nmely, unit irulr-r grphs n proper irulr-r grphs in [13], horl proper irulr-r grphs in [1], oiprtite irulrr grphs in [12] n lter in [6] (using so-lle ege-sterois), n Helly irulr-r grphs within irulr-r grphs in [11] (using so-lle ostles). More reently, in [2], the uthors gve forien inue sugrph hrteriztions for P 4 -free irulr-r grphs, imon-free irulr-r grphs, pw-free irulr-r grphs, n most relevnt for this pper, they hrterize lw-free horl irulr-r grphs. Our results (nmely Theorem 3.2) my e seen s omplementing their work, sine in this regr we give forien inue sugrph hrteriztion of K 5 -free horl irulr-r grphs. 1 MF prtilly supporte y the grnt ANR-09-JCJC-0041. E-mil: mthew@ims.res.in 2 PH prtilly supporte y the uthor s NSERC Disovery Grnt. E-mil: pvol@sfu. 3 JS grtefully knowleges support from EPSRC, grnt EP/I01795X/1. E-mil: j.stho@wrwik..uk
2 Bloking quruple To uil intuition, we strt y relling the efinition of steroil triple. We sy tht vertex x misses pth P in G if x hs no neighour on P. Verties x, y, z form n steroil triple of G if etween ny two of them, there is pth in G misse y the thir vertex. It is esy to see tht n intervl grph nnot hve n steroil triple [10]. We sy tht verties x, y voi verties z, w in G if there exists n xy-pth misse y oth z n w, or there exists zw-pth misse y oth x n y. We sy tht verties x, y, z, w form loking quruple (BQ) of G if ny two of them voi the remining two. Lemm 2.1 If G is irulr-r grph, then G hs no loking quruple. To see this, oserve tht BQ is lwys n inepenent set of size four. Now, suppose tht G hs irulr-r representtion n the rs representing verties x, y, z, w pper in this irulr orer. Then no pth etween x n z n e misse y oth y n w, n no pth etween y n w n e misse y x n z. In other wors, the verties x, z o not voi y, w. Let us now isuss vrious forms of loking quruples tht one my enounter in grphs. One lss of suh exmples rises from steroil triples: if,, form n steroil triple of G n is vertex of egree zero in G, then,,, is loking quruple. This n e seen in the first three grphs in Figure 1. Other wys of extening n steroil triple to BQ re lso illustrte in Figure 1. The verties,, in eh of the grphs in the seon row form n steroil triple while the verties,,, form loking quruple. For horl grphs, these re ll possile forms of BQs (see Theorem 3.1). Unlike these exmples, the two horl grphs in Figure 2 o not ontin loking quruples, n yet they re not irulr-r grphs. Thus the sene of loking quruples is not suffiient to gurntee tht (horl) }{{... } 2 eges... Fig. 1. Forien inue sugrph hrteriztion of horl grphs with no BQs.
Fig. 2. Some miniml horl non-irulr-r grphs with no BQs. grph is irulr-r grph. However, in some ses, it my e suffiient. For instne, result of [2] (Corollry 15) n e restte s follows. Lemm 2.2 A lw-free horl grph is irulr-r grph iff it hs no BQ. We prove similr sttement for horl grphs of inepenene numer t most four (see Theorem 3.2). The sene of BQs therefore gives us simple n uniform forien struture hrteriztion of these lsses, s oppose to more ommon forien inue sugrph hrteriztions [1,2,11,13]. 3 Min results In this setion, we summrize the min theorems of this pper. Firstly, we esrie ll miniml forien inue sugrphs hrterizing horl grphs with no BQs. These re the grphs epite in Figure 1. Theorem 3.1 If G is horl, then the following re equivlent. (i) G ontins loking quruple. (ii) G ontins n inue sugrph isomorphi to grph in Figure 1. In ft, the theorem hols for the more generl lss of nerly horl grphs ( grph lss efine in [3] generlizing oth horl n irulr-r grphs). Seonly, we show tht the sene of BQs is neessry n suffiient for horl grph of inepenene numer α(g) 4 to e irulr-r grph. Theorem 3.2 If G is horl n α(g) 4, the following re equivlent. (i) G is irulr-r grph. (ii) G ontins no loking quruple. The theorem fils for horl grphs G with α(g) 5 s Figure 2 shows. 4 Proof skethes To prove Theorem 3.1, onsier horl grph G with loking quruple,,,. By symmetry, we my ssume tht G ontins n -pth P misse y oth n, n n -pth P misse y oth n. If G lso ontins
l e n o h i j p m 0 1 2 i,n,i,p,o h,i,m i,m,p 19 16 17 18 h,l,o h,i,o h,i,j 3 4,l,o e,h,h,j 5 15 11 10 14 6 7 13 9 8 12 o 1 18 19 0 2 17 3 16 4 15 5 14 6 13 7 12 8 11 10 9 j m ) ) ) n i p l e h v φ(v) l v r v 8 7 9 2 1 3 0 19 1 13 12 14 e 5 4 6 h 6 2 12 i 15 8 19 j 8 6 10 l 2 0 4 m 12 10 13 n 16 15 17 o 1 17 5 p 13 11 15 Fig. 3. ) Exmple horl grph, ) lique tree + n Euler tour, ) resulting rs -pth misse y n, then,, is n steroil triple in G N[]. Thus y [10], G ontins one of the grphs in the top row of Figure 1, or one of the first two grphs in the seon row of Figure 1 (ignoring the lel ). If this is not the se, then G ontins n -pth P misse y oth n. We hoose,,, so s to minimize P + P + P. It n e shown tht this minimlity omine with horlity of G implies tht the union of verties of the three pths P, P, P inues in G extly one of the grphs in the seon row of Figure 1. This is rther tehnil, so we omit further etils. For Theorem 3.2, let G e horl grph. The iretion (i) (ii) is prove s Lemm 2.1. For (ii) (i), ssume (ii). If α(g) 2, then G ontins no steroil triple. So G is n intervl grph y [10] whih implies (i). Suppose tht α(g) = 3. Note tht (ii) is vuously stisfie in this se. Let T e lique tree of G (sutree intersetion moel, see [7]). Sine G is perfet grph, it hs lique over y three liques. Let Q 1, Q 2, Q 3 e these liques. This implies tht T hs t most three leves, n eh lef of T is one of Q 1, Q 2, Q 3. If T hs two leves, then G is n intervl grph implying (i). So we my ssume tht T hs extly three leves, nmely Q 1, Q 2, Q 3. From here, we proee s follows. We fix some Euler tour A 0, A 1,..., A k 1 of T (onsiere s igrph) where A i re noes of T. Using this tour, we onstrut irulr-r representtion of G. We strt y pling k points λ 0, λ 1,..., λ k 1 on the irle rrnge in this orer in the lokwise iretion. Then for every vertex v of G we onstrut irulr-r s follows. Rell tht v Q i for some i {1, 2, 3}, sine Q 1, Q 2, Q 3 is lique over of G. We hoose i suh tht v Q i n then hoose in the Euler tour some ourrene of Q i. Nmely, we hoose φ(v) suh tht A φ(v) = Q i. Then we wlk from A φ(v) long the tour (in oth iretions) s fr s possile so long s the liques we enounter ontin v. Tht is, we efine inies l v, r v suh tht l v = φ(v) min{i 0 v A φ(v) i } r v = φ(v) + min{j 0 v A φ(v)+j }
We represent v y lokwise r from λ lv+1 to λ rv 1 (inies tken moulo k). We refer to Figure 3 for n illustrtion of this proess. It n e shown tht in se α(g) = 3 this onstruts vli irulr-r representtion of G. So we my ssume tht α(g) = 4. We proee similrly s in the previous se ut with more re. Let T e lique tree of G. Sine G is perfet, T hs lique over y four liques. Let Q 1, Q 2, Q 3, Q 4 e these liques. It follows tht T hs t most four leves n eh lef of T is one of Q 1, Q 2, Q 3, Q 4. Assume first tht T hs extly four leves, nmely Q 1, Q 2, Q 3, Q 4. Sine T is lique tree, for eh i {1, 2, 3, 4}, there exists vertex v i tht elongs to Q i n no Q j, j i. Let H enote the grph on verties {v 1, v 2, v 3, v 4 } where v i v j is n ege if n only if v i, v j voi verties V (H)\{v i, v j }. We use H to guie us in the next steps. We hoose n Euler tour of T stisfying ( ): ( ) if v i v j E(H) n {v i, v j } = V (H) \ {v i, v j }, then the sutour of the Euler tour etween Q i n Q j ontins neither or oth of Q i, Q j. It n e shown tht this is lwys possile for some lique tree T hving Q 1, Q 2, Q 3, Q 4 s leves. For this, we use the ft tht H is either empty, or 4-yle, or pir of isjoint eges, euse v 1, v 2, v 3, v 4 is not loking quruple of G y (ii). From this point we proee s in the se α(g) = 3. Finlly, if T hs three leves or less, we gin hve pirwise non-jent verties v 1, v 2, v 3, v 4 in the liques Q 1, Q 2, Q 3, Q 4, n we efine the grph H s efore. We use H to efine φ (rther thn to hoose n Euler tour) using whih we onstrut irulr-rs s in the se α(g) = 3. We omit further etils. Referenes [1] Bng-Jensen, J. n P. Hell, On the horl proper irulr-r grphs, Disrete Mthemtis 128 (1994), pp. 395 398. [2] Bonomo, F., G. Durán, L. N. Grippo n M. D. Sfe, Prtil hrteriztions of irulr-r grphs, Journl of Grph Theory 61 (2009), pp. 289 306. [3] Brnstät, A., n C. T. Hoàng, On lique seprtors, nerly horl grphs, n the Mximum Weight Stle Set prolem, Theoretil Computer Siene 389 (2007), pp. 295 306. [4] Booth, K. S. n G. S. Lueker, Testing for the onseutive ones property, intervl grphs n grph plnrity using PQ-tree lgorithms, Journl of Computer n System Sienes 13 (1976), pp. 335 379. [5] Corneil, D. G., S. Olriu n L. Stewrt, The LBFS struture n reognition
of intervl grphs, SIAM Journl on Disrete Mthemtis 23 (2009), pp. 1905 1953. [6] Feer, T., P. Hell n J. Hung, List homomorphisms n irulr-r grphs, Comintori 19 (1999), pp. 487 505. [7] Gvril, F., The intersetion grphs of sutrees in trees re extly the horl grphs, Journl of Comintoril Theory B 16 (1974), pp. 47 56. [8] Gilmore, P. C. n A. J. Hoffmn, A hrteriztion of omprility grphs n of intervl grphs, Cnin Journl of Mthemtis 16 (1964), pp. 539 548. [9] Klee, V., Wht re the intersetion grphs of rs in irle?, Amerin Mthemtil Monthly 76 (1976), pp. 810 813. [10] Lekkerkerker, C. G. n J. C. Boln, Representtion of finite grph y set of intervls on the rel line, Funment Mthemtie 51 (1962), pp. 45 64. [11] Lin, M. C. n J. L. Szwrfiter, Chrteriztions n liner time reognition of Helly irulr-r grphs, in: COCOON 2006, Leture Notes in Computer Siene 4112/2006, pp. 73 82. [12] Trotter, W. T. n J. I. Moore, Chrteriztions prolems for grphs, prtilly orere sets, ltties n fmilies of sets, Disrete Mthemtis 16 (1976), pp. 361 381. [13] Tuker, A., Struture theorems for some irulr-r grphs, Disrete Mthemtis 7 (1974), pp. 167 195.