MENGERIAN THEOREMS FOR PATHS OF BOUNDED LENGTH

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1 Periodica Mathematica Hugarica Vol. 9 (4), (1978), pp MENGERIAN THEOREMS FOR PATHS OF BOUNDED LENGTH by L. LOVASZ (Szeged), V. NEUMANN-LARA (Mexico) ad M. PLUMMER (Nashville) Dedicated to the memory of FERNANDO EscALANTE 1. Itroductio Let u ad v be o-adjacet poits i a coected graph G. A classical result kow to all graph theorists is that called MENGER's theorem. The poit versio of this result says that the maximum umber of poit-disjoit paths joiig u ad v is equal to the miimum umber of poits whose deletio destroys all paths joiig u ad v. The theorem may be proved purely i the laguage of graphs (probably the best kow proof is idirect, ad is due to DiRAC [3] while a more eglected, but direct, proof may be foud i ORE [7]). Oe may also prove the theorem by appealig to flow theory (e.g. BERGE [1], p. 167). I may real-world situatios which ca be modeled by graphs certai paths joiig two o-adjacet poits may well exist, but may prove essetially useless because they are too log. Such cosideratios led the authors study the followig two parameters. Let be ay positive iteger ad let u ad v be ay two o-adjacet poits i a graph G. Deote by A (u, v) the maximum umber of poit-disjoit paths joiig u ad v whose legth (i.e., umber of lies) does ot exceed. Aalogously, let V (u, v) be the miimum umber of poits i G the deletio of which destroys all paths joiig u ad v which do ot exceed i legth., A special case would obtai whe = p = I V(G)I, ad we have by Moger's theorem, the equality A (u, v) = V,,(u, v). to Fig. I Research supported i part by CIMAS (The Uiversity of Mexico), IREX ad The Hugaria Academy of Scieces. AMS (MOS) subject classificatios (1970). Primary 05C35. Key words ad phrases. Meger's theorem, disjoit paths, miimum cut-sets.

2 270 LOVASZ, NEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH I geeral, however, oe does ot have equality, but it is trivial that A (u, v) :!E~ V (u, v) for ay positive iteger. O the other had, the graph of Fig. 1 has VS (u, v) = 2, but A S (u, v) = 1. We prefer to formulate our work as a study of the ratio V(u, v) or A (u, v) simply A whe the poits u ad v are uderstood. For ay termiology ot defied i this paper, the reader is referred to the book by HARA Y [4]. 2. Bouds for the ratio As i the itroductio we shall assume throughout this paper that u ad v are o-adjacet poits i the same compoet of a graph G. It is trivial that 1 < V (u, v) < - 1. As usual, d(u, v) deotes the distace A(u, v) betwee poits u ad v. Our first result ivolves this distace. THEOREM 1. For every positive iteger > 2 ad/or each m = - d(u, v) > moo, V(u,v) <m+ 1. A (u, v) - The costructio i Sectio 3 shows that this boud is sharp. PROOF. The proof proceeds by iductio o m. Hece first let m 0, i.e., suppose = d(u, v) = o. We oriet some of the lies of G accordig to the followig rule : let xy be ay lie. The if d(x, v) > d(y, v), oriet x to y. The, clearly, ay u-v geodesic (i.e., a shortest u-v path) yields a dipath from u to v. O the other had, we claim that ay u-v dipath must arise from a geodesic u-v path i G, for just cosider our rule of orietatio. If (x, y) is a directed lie i our dipath, d(x, v) > d(y, v) ad distace decreases by 1 as we traverse each dilie toward v. Hece our dipath caot have > lies ad hece must have come from a u-v geodesic. Thus i the orieted subgraph of G, the u-v paths are exactly the geodesics, so by Meger's theorem,,v (u, v) = A (u,. :v) ad the case for m = 0 is proved.. Now by iductio hypothesis, assume that the theorem holds for some mq > 0 ad suppose m = - d(u, v) 'm o + 1 (ad hece that > d(u, v)). Let X be a miimum set of poits coverig all u-v geodesics. By the case for m = 0, X I = Vd(U,)(u, v) = Ad(U,ro)( u, V) S A (u, V). Cosider the graph G - X. If dg_ x (u, v) >, X has covered all u-v paths of legth < ad we have, V,,(u, v) = I X I < A (u, v) < ma(u, v) ad we

3 LOVASZ, NEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH 271 doe. So suppose dg_ x (u, v) <, say dg_ x (u, v) = - t for some t, t < m. (Note that t < m for X destroys all u-v geodesics ad thus - dg-x(u, v) < - d(u, v) = m). So by the iductio hypothesis applied to poits u ad v i graph G-X, have V -x (u, v) < (t + 1) A-x (u, v). it we ca the cover all -paths i G joiig u ad v with a set Y where I Y I = IXI + ( t + 1) A-"(u, V) < I X I + (t + 1) A (u, v). V(u, V) < I X I + (t + 1) A(u, v) < (t + 2) A(u, v) < (m + 1)A(u, v) id the proof is complete. The ext theorem shows that we ca do better as far as a boud deped-.g solely upo is cocered. THEOREM 2. For ay graph 0, ay o-egative iteger, ad ay two o-adjacet poits u ad v, V(u, v) < [_JA(u,V). PRooF. If d(u, v), ~ /2 + 1, we are doe by Theorem 1. So suppose (u, v) < ( + 1)/2. Choose D such that d(u, v) < D < ad let P0 be a s-v geodesic i G. Form a ew graph-- G, from G ' by "removig all iterior poits of P 0. Clearly d0l (u, v) Z dg(u, v).. Now remove ay u-v geodesic G1, say P1, to obtai G2. Cotiue i this maer util we obtai a graph a-, cotaiig a u-v geodesic P, such that l(p,) < D, but the legth of ay c-v geodesic i G, +1 > D. For coveiece let us deote G, +1 by G' ad similarly for parameters of this graph. Thus dg, +1 (u, v) = d'(u, v) > D + 1. Sice we have removed r disjoit u-v paths from G to get G', we have AZA+r, ( 1 ) for all discarded paths had legth o greater tha the legth of a u-v geodesic i G'. Also V < V + r(d - 1). (2) Moreover, if G' is coebted, we have by Theorem 1 that V<(-d'(u,v)+1)A (-D-1~-1)A (-D)A. The combiig (2) ad (3), we obtai by (1) V< ( - D)A' + r(d - 1) < ( - D)(A,, -r)+r(d-1)= =(-D)A+r(2D--1). (3)

4 272 LOVASZ, NEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH Sice r is o-egative, choose D to be the greatest iteger so that 2D-- 1<0. Hece D S + 1 J ad Sice D is itegral, D= 2 1 J. Hece - D= - C 2 1 J= C2 J ad thus V,, S [ 2 ]A. If 0' is ot coected betwee u ad v, we have A = V = 0 ad coclude similarly. The boud i this theorem is sharp for = 2, 3 ad 5 (for = 5, see Fig. 1). It is, however, ot sharp for = 4. THEOREM 3. For ay graph 0 with o-adjacet Paits wad v, V,(u, v) A4(u, v). PROOF. Partitio the poits of 0 - u - v ito disjoit classes (i, j) as follows : w E (i, j) iff d(u, w) = i ad d(w, v) = j. Clearly we may igore classes (1, 1) ad all (i, j) for i + j > 4. So the remaiig graph Q has the appearace of Figure 2. s It it A. U "4K$AI Fig. 2 Now costruct a di-graph b as follows. Let V(b) = V(O) ad (x, y) E E(b) _iff (1) xy E E(O) ad (2) d(u, y) > d(u, x). Hece b has the appearace of Figure 3. Fig. 3

5 LOVASZ, NEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH 273 Observe that (a) each dipath i b has _legth S 4 ad _ (b) each chordless path of G of legth S 4 correspods to a dipath i D. Let S be a set of V4 poits i G - u - v whose deletio destroys all u-v paths of legth S 4. But the i b - u - v all dipaths from u to v are also destroyed, so V4 > H(u, v) where H(u, v) deotes the miimum umber of poits whose deletio separates u ad v i D. But by Meger's theorem applied to b, H(u, v) (= the maximum umber of poit-disjoit dipaths from u to v) S A4, sice each set of poit-disjoit _dipaths from u to v i b correspods to a set of poit-disjoit u-v paths i Q of the same cardiality. Thus it will suffice to prove V4 S H(u, v). Let L be ay set of H(u, v) poits i D - u - v whose removal separates u ad v. We ow claim L meets all u-v paths i 0 of legth S 4. If ot, there is a path P joiig u ad v with legth :< 4 ad (V(P) - u - v) fl L = 0. We may assume P is chordless. But, the it traslates ito a dipath from u to v i b o the same poits. L does ot meet this dipath, which is a cotradictio. I the costructio of the ext sectio we will have - = 2 or f I/ It is ukow to us where for a fixed, the value of sup lies Ly i the iterval ft 21, [2]). 3. A Costructio We will costruct a graph G(, t) such that give t(> 0), there is a ad a graph G(, t) which has 2 distict o-adjacet poits u ad v such that A (u, v) = 1, but V (u, v) = t + 1. Moreover, we will show i additio that give ay iteger k(> 1), we ca costruct a G(, t, k), which is k- coected. For the momet, suppose t is a give positive iteger. Choose ay > t + 1 ad fix it. Costruct a, path L of legth s = - t joiig u ad v. As is customary, we shall refer to paths havig at most their edpoits i commo as opely disjoit. Now for each i, 2 S i S t + 1, take every pair of poits a, b o L which are at a distace = i o L ad attach a path of legth i + 1 at a ad b which is opely disjoit from L. Such paths we shall call ears. (See Figure 4). Now let P be ay u-v path of legth = s'(s ). P has at least - t lies sice L is a u-v geodesic. Suppose P uses r ears. Sice replacig a ear by the correspodig segmet of L shortes the legth by > 1, we have s'> - t + r. Hece 2 Periodica Math. 9 (4)

6 274 LOVASZ, NEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH... o--ov legth (L)=s=-t Fig. 4 r < t. Sice each ear has S t + 1 iterior poits, P has S r(t + 1) poits ot o L. So the umber of poits of P o L - is (ot icludig u ad v) >(s'-1)-r(t+1)z-t+r-1-r(t+1)= =-(r+1)t-1>-(t+1)t-1. If - (t + 1) t - 1 > 2 (the umber of ier poits of L), the ay two such paths P must have a iterior poit i commo. Note that the umber of ier poits of L = - t - 1. Thus what we eed is that (t + 1) t - 1 I ( - t - 1), i.e., Z 2t2 + t + 2. If is give, the best t satisfyig this iequality is either L 2-1 or C 2 J. The with such a, ay two u-v paths of legth S must have some ier poit of L i commo ; i.e., A,,(u, v) = 1. We ow proceed to show that V,,(u, v) Z t + 1. Suppose there is a set T of t poits which cover all u-v paths of legth S. We may assume all poits of T lie o L, for otherwise move right o the "offedig ear" util L is reached ad use the poit of L thus ecoutered i place of the origial T-poit. If the ear eds at v take the left-had ed poit o L. Note also that u, v are joied by o oe ear by our choice of. Let us call the sets of poits of T which are cosecutive o L the blocks of T. There are o more tha t such blocks. Recall that L cotais - t + 1 poits where - t + 1 = ( + 1) - t > 3 ad hece - t Z 2. Thus we ca form a ew u-v path Q by jumpig each block of T with a ear. This ew path Q the misses T ad we have added exactly oe to the legth of L for each block jumped. It follows that Q has legth < s + t = - t + t =. Hece, there is a u-v path Q of legth S which misses T cotradictig the defiitio of T. Thus V (u, v) > t + 1.

7 LOVASZ, NEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH 275 We kow at this poit that G(, t) is at least 2-coected. Let k be ay iteger ~ 2. We ow proceed to modify the graph G(, t) costructed above so that the resultig graph G(, t, k) retais the properties that A (u, v) = 1, V > t + 1 ad i additio is k-coected. The idea is to costruct a ew graph H, joi it to G(, t) by suitably chose lies so that the resultig graph is k-coected, but also so that o ew "short" u-v paths are itroduced. Let the poits of G(, t) be w1,..., WN. Further, let M = k +. Form a path of MN poits plp2.. PMN ad the replace each p; with a clique, Kk, o k poits where each poit of Kk is joied to each poit of Kk+ l. Now joi wl to exactly oe poit of each of Kk,..., Kk ; w2 to exactly oe poit of K,M +1,, K M+k ; ad, i geeral, w1 to exactly oe poit of K(kJ-1)M+i,... Kkj-1>M+k for j = 1,..., N. It is ow easily see that o ew path joiig ay w; ad wj is of legth < + 1. It is clear that A = 1 ad V = t + 1 i this ew graph for ay path of legth < joiig u ad v must lie etirely withi the origial G(, t) part of this ew graph. It is equally clear that the ew graph G(, t, k) is k-coected. 4. A differet type of Megeria result I this sectio we take a differet approach. Recall that V (u, v) > A (u, v) ad moreover, strict iequality ca occur. Oe's ituitio may idicate that eve i this case, if the subscript o A is allowed to icrease to some ew value ' oe ca always obtai V < A,,,. The ext theorem says that such a cojecture is ot oly appealig, but true. THEOREM 4. Let ad h be positive itegers. The there is f (, h) such that if V,, (u, v) Z h, the Af(,,,h) (u, v) > h. I the proof we eed the followig result. a costat THEOREM 5 (BoLLOBks [2], KATONA [6], JAEGFER^-PAYAN [5]). Give ay family o f r-sets which eeds at least t poits to cover, the there exists a sub family, - with < r + t 1 elemets which still eeds t poits to cover. r REMARK. It is trivial to see that istead of "r-sets" oe ca say "sets of size at most r". PROOF of Theorem 4. Cosider sets of iterior poits of u-v paths of legth <. By the assumptio we eed > h poits to cover the members of this family. By the precedig theorem ad the remark followig it we ca select (+h-21 paths of legth < such that we still eed h poits to cover these l -1 J 2*

8 276 LOVASZ, 1tEUMANN-LARA, PLUMMER : PATHS OF BOUNDED LENGTH paths. So let Gl be the uio of these paths ad apply Meger's theorem to Gl to see that there are > h opely disjoit u-v paths. So how log ca a h-2 logest path i G 1 be? We have + paths of legth <. -1 h- + So Gl - u - v has < ( - 1) 2 ) poits. Now amog all sets of -1 Z h opely disjoit u-v paths i G1, the logest path oe could fid would be + h - of legth ( - 1) 2 - (h - 1)+ 1. (This of course happes whe oe -1 has h - 1 paths of legth 2 ad a sigle log path of the above legth.) + h - 2 Thus set f (, h) = ( - 1) h + 2 ad we have Af(,, h)(u, v) > h. N -1 ~- REFERENCES [1] C. BERGE, Graph8 ad hypergraph8, North-Hollad, Amsterdam, MR 50 # 9640 [2) B. BOLLOBAs, O geeralized graphs, Acta Math. Acad. Sci. Hugar. 16 (1965), MR 32 * 1133 [3] G. Dia.&c, Short proof of Meger's graph theorem, Mathematika 13 (1966), MR 33 # 3956 [4] F. HARARY, Graph theory, Addiso-Wesley, Readig, MR 41 # 1566 [5] F. JAEGER ad C. PAYAN, Nombre maximal d'aretes d'u hypergraphe r-critique de rag h, C. R. Acad. Sci. Pari8. Ser. A 273 (1971), Zbl [6] G. KATONA, Solutio of a problem of A. Ehrefeucht ad J. Mycielski, J. Combiatorial Theory Ser. A 17 (1974), MR 49 * 8870 [7] O. ORE, Theory of graph8, Amer. Math. Soc. Colloq. Publ., Providece, MR 27 * 740 JOZSEF ATTILA TUDOMANYEGYETEM BOLYAIINTIZET H-6720 SZEGED ARADI VERTANIIK TERE 1. HUNGARY (Received November 18, 1975) INSTITUTO DE MATEMATICAS UNIVERSIDAD NACIONAL AUTONOMA DE MEXICO VILLA OBREGON CIUDAD UNIVERSITARIA MEXICO 20. D. F. MEXICO DEPARTMENT OF MATHEMATICS VANDERBILT UNIVERSITY NASHVILLE, TN U. S. A.