RELATIONS BETWEEN MEDIAN GRAPHS, SEMI-MEDIAN GRAPHS AND PARTIAL CUBES

Univrsity of Ljubljana Institut of Mathmatics, Physics an Mchanics Dpartmnt of Mathmatics Jaranska 19, 1000 Ljubljana, Slovnia Prprint sris, Vol. 36 (1998), 612 RELATIONS BETWEEN MEDIAN GRAPHS, SEMI-MEDIAN GRAPHS AND PARTIAL CUBES Wilfri Imrich Hnry Martyn Mulr Sani Klavžar Rist Škrkovski ISSN 1318-4865 Jun 16, 1998 Ljubljana, Jun 16, 1998

Rlations btwn mian graphs, smi-mian graphs an partial cubs Wilfri Imrich Dpartmnt of Appli Mathmatics, Montanunivrsitat Lobn Franz-Josf Stra 18, A-8700 Lobn, Austria Sani Klavzar Dpartmnt of Mathmatics, PEF, Univrsity of Maribor Koroska csta 160, 2000 Maribor, Slovnia Hnry Martyn Mulr y Economtrisch Instituut, Erasmus Univrsitit P.O. Box 1738, 3000 DR Rottram, Th Nthrlans Rist Skrkovski z Dpartmnt of Mathmatics, Univrsity of Ljubljana Jaranska 19, 1111 Ljubljana, Slovnia Jun 16, 1998 Abstract Arich structur thory has bn vlop in th last thr cas for graphs mbabl into hyprcubs, in particular for isomtric subgraphs of hyprcubs, also known as partial cubs, an for mian graphs. Mian graphs, which constitut a propr subclass of partial cubs, hav attract consirabl attntion. In orr to bttr unrstan thm smi-mian graphs hav bn introuc but hav turn out to form a rathr intrsting class of graphs by thmslvs. This class lis strictly btwn mian graphs an partial cubs an satiss an xpansion thorm for which wgiv a slf-contain proof. Furthrmor, w show that ths graphs can b charactriz as til partial cubs an w prov that for a smi-mian graph G with n vrtics, m gs an k quivalnc classs of Djokovic's rlation, w hav 2n;m;k 2 : Morovr, quality hols if an only if G contains no K 2 2C 2t with t 2 as a subgraph. For mian graphs w show that thy can b charactriz as smi-mian graphs which contain no convx Q ; 3, i.. th 3-cub minus a vrtx, as a convx subgraph. Also, w introuc th concpt of wak 2-convxity anus it, among othr things, to prov that mian graphs ar bipartit, msh graphs with wakly 2-convx intrvals. Support by th Ministry of Scinc an Tchnology of Slovnia unr th grant J1-0498-0101 an by th SWON, th Nthrlans. y Support by th Ministry of Scinc an Tchnology of Slovnia an by th NUFFIC, th Nthrlans. z Support by th Ministry of Scinc an Tchnology of Slovnia unr th grant J1-0502-0101. 1

1 Introuction A partial cub is a connct graph that amits an isomtric mbing into a hyprcub. Th structur of partial cubs is rlativly wll unrstoo. Lt us just mntion hr thr funamntal rsults. First, Djokovic [7] charactriz ths graphs via convxity of crtain vrtx partitions. H also introuc a rlation, which was latr us by Winklr [24] to charactriz partial cubs as thos bipartit graphs for which is transitiv. As th thir rsult w mntion an xpansion thorm of Chpoi [3]. Partial cubs hav foun svral application, s, for instanc, arly rfrncs [11, 12] rlat to communication thory an rcnt applications [6, 15] to chmical graph thory. (For cint computation of s [1, 8].) A mian graph is a connct graph such that, for vry tripl of its vrtics, thr is a uniqu vrtx lying on a gosic (i.. shortst path) btwn ach pair of th tripl. By now, th class of mian graphs has bn wll invstigat an a rich structur thory is availabl, s.g. [2, 5, 17, 23]. Although mian graphs form a propr subclass of partial cubs, no nontrivial charactrization of mian graphs as partial cubs fullling som aitional conitions is known. In orr to bttr unrstan mian graphs an thir rcognition complxity, smimian graphs wr introuc in [13] as partial cubs satisfying a crtain aitional conition to b xplain in Sction 2. In th sam papr smi-mian graphs ar charactriz in a similar way as Winklr charactriz partial cubs in [24], an an xpansion thorm for smi-mian graphs was stablish. This thorm is similar to Mulr's xpansion thorm for mian graphs [18, 19] an to th on of Chpoi for partial cubs [3]. For furthr rsults on isomtric mbing an in particular on partial cubs, w rfr to [9, 10, 22]. In this papr w stuy intrrlations btwn th abov mntion classs of graphs. In th nxt sction w introuc smi-mian graphs an prsnt rsults concrning Djokovic's rlation which will b us in th squl. In Sction 3 w stuy xpansion procurs for graphs an giv a irct, slf-contain proof of th xpansion thorm for smi-mian graphs. This rsult was rst stablish in [13] with a proof bas on Chpoi's xpansion thorm for partial cubs from [3]. Morovr, as svral rsults of this papr rly on th xpansion for smi-mian graphs, w bliv that th nw tail proof is justi. In Sction 4 w introuc th concpt of wak 2-convxity an prov that mian graphs ar prcisly smi-mian graphs without convx Q ; 3. Thn w stablish a connction btwn partial cubs an smi-mian graphs by charactrizing smi-mian graphs as til partial cubs. Ths two rsults ar thn combin to scrib mian graphs as thos til partial cubs which contain no convx Q ;. 3 In Sction 6 w prov an Eulr-typ inquality for smi-mian graphs an show that th quality occurs if an only if th smi-mian graph consir contains no K 2 2C 2t with t 2 as a subgraph. This rsult is thn us to stuy th numbr of facs of a planar mian graph. W conclu th papr with two opn problms. For an arbitrary rlation w will writ for its transitiv closur. Q ; 3 stans for th graph obtain from th 3-cub by rmoving on of its vrtics. For a graph G =(V E) an X V lt hxi not th subgraph inuc by X. For u v 2 V (G) lt G (u v) not th lngth of a shortst path in G from u to v. A subgraph H ofagraphg is an isomtric subgraph, if H (u v) = G (u v) forallu v 2 V (H). 2

Th intrval I(u v) btwn vrtics u an v consists of all vrtics on shortst paths btwn u an v. A subgraph H of G is convx, if for any u v 2 V (H), w hav I(u v) V (H). Clarly, a convx subgraph is connct. Th Cartsian prouct G2H of graphs G an H is th graph with vrtx st V (G)V (H) an (a x)(b y) 2 E(G2H) whnvr ab 2 E(G) an x = y, or, if a = b an xy 2 E(H). Th n-cub Q n is th Cartsian prouct of n copis of th complt graph on two vrtics K 2. 2 Th rlation an smi-mian graphs Th Djokovic's rlation introuc in [7] is n on th g-st of a graph in th following way. Two gs = xy an f = uv of a graph G ar in rlation if G (x u)+ G (y v) 6= G (x v)+ G (y u) : Clarly, is rxiv an symmtric. If G is bipartit, thn can b n as follows: = xy an f = uv ar in rlation if (x u) =(y v) an (x v) =(y u) : Among bipartit graphs, is transitiv prcisly for partial cubs (i.. isomtric subgraphs of hyprcubs), as has bn prov by Winklr in [24]. Sinc mian graphs form a subclass of partial cubs, is transitiv in particular for mian graphs. In othr wors, is a congrunc on mian graphs. Lt G =(V E) b a connct, bipartit graph. For any g ab of G w writ W ab = fw 2 V j G (a w) < G (b w)g, U ab = fw 2 W ab j w has a nighbor in W ba g, F ab = f 2 E j is an g btwn W ab an W ba g : Not that, G bing bipartit, w hav V = W ab [ W ba. For any two gs uv an xy that ar in rlation, w writ uv==xy if (u x) =(v y) = (u y) ; 1 = (v x) ; 1. A st of gs is call -transitiv if any two gs in F ar in rlation. Lmma 2.1 Lt G b aconnct, bipartit graph, an lt ab b any g of G. Thn F ab is th st of all gs in rlation with ab. Proof. Lt uv b any g in F ab with u in U ab an v in U ba. Thn, by th nitions of W ab an U ab,whav (a u) =(b u) ; 1 (b v) =(a v) ; 1 (a u) : So uv an ab ar in rlation. Sinc both ns of any g in W ab ar closr to a than to b, no g in W ab can b in rlation with ab. 2 Thorm 2.2 Lt G b a connct, bipartit graph, an lt ab b an g of G. Thn th st F ab is -transitiv if an only if W ab an W ba ar convx in G. 3

Proof. First, lt both W ab an W ba b convx in G. Tak any two gs uv an xy in F ab with u x in U ab an v y in U ba. Thn nithr v nor y is on a u x-gosic, whnc (u x) = (u y) ; 1=(x v) ; 1. Similarly, nithr u nor x is on a v y-gosic, whnc (v y) =(u y) ; 1=(x v) ; 1. This implis that uv==xy. So F ab is -transitiv. Convrsly, lt F ab b -transitiv. First w claim that, for any gs uv an xy in F ab with ab==uv an ab==xy, w hav uv==xy. Assum th contrary, an lt uv an xy b such that (u y) is as small as possibl. Not that now w must hav uv==yx. Lt P b a u v- gosic. Sinc u is in W ab an y is in W ba, thr must b an g pq on P, going from u to y, suchthatp is in W ab an q is in W ba. So pq is in F ab. By th minimalityof(u y), w hav pq==xy, so that (p x) = (q y) = (p y) ; 1. But this is impossibl, sinc q is btwn p an y on P, an th subpath of P btwn p an y is a p y-gosic. This provs th claim. Nxt, choos any two vrtics u an v in W ab. Lt P b any u v-path that passs through W ba. Thn P must contain at last two gs of F ab. Going from u to v along P, lt xy b th rst an qp b th last g in F ab,say, with ab==xy an ab==pq. Thn w hav xy==pq, so that (x p) = (x q) +1. This implis that th subpath of P btwn x an p is not a gosic (sinc it contains q). Hnc w conclu that ach u v-gosic lis compltly in W ab. Similarly, W ba is convx. 2 Th nxt corollary is an immiat consqunc of Thorm 2.2. Corollary 2.3 Lt G b a connct, bipartit graph, an lt ab b an g of G. If F ab is -transitiv, thn F ab is a matching an inucs an isomorphism btwn hu ab i an hu ba i. A bipartit graph G is a smi-mian graph if = an th subgraph hu ab i is connct for any gab of G. Equivalntly, G is a smi-mian graph if G is a partial cub an if all th subgraphs hu ab i ar connct. Sinc in mian graphs th subgraphs hu ab i ar convx, mian graphs form a propr subclass of smi-mian graphs. Two typical smi-mian graphs ar givn in Fig. 1. (Consir two vrtics at istanc 4 from th outr 8-cycl of th scon graph to not that C 8 canbanintrval of a smi-mian graph.) u Figur 1: Smi-mian graphs 4

Smi-mian graphs ar charactriz in [13] as bipartit graphs for which = hols. Hr an g is in rlation to an g f if an f ar opposit gs of a squar in G or if = f. 3 Expansions, contractions an an xpansion thorm In [21] a broa schm for xpansions an contractions in graphs was prsnt. Hr w focus on on spcic instanc, th cas of smi-mian graphs. As th main rsult of this sction w prov an xpansion thorm for smi-mian graphs. This rsult has for th rst tim bn givn in [13] with a proof which rlis on th Chpoi's xpansion thorm for partial cubs, s [3, 4]. Hr w prsnt a irct, slf-contain proof of it. As a by-prouct w obtain svral rsults which might b of inpnnt intrst. Th notations w vlop in this sction will b us in th squl without furthr xplanation. Lt G =(V E) b a connct graph. For two subgraphs G 1 =(V 1 E 1 ) an G 2 =(V 2 E 2 ) of G, thintrsction G 1 \G 2 is th subgraph of G with vrtx-st V 1 \V 2 an g-st E 1 \E 2, an th union G 1 [ G 2 is th subgraph of G with vrtx-st V 1 [ V 2 an g-st E 1 [ E 2. A covr G 1, G 2 of G consists of two subgraphs G 1 an G 2 of G such that G 0 = G 1 \ G 2 is non-mpty ang = G 1 [ G 2. Not that thr ar no gs btwn G 1 ; G 2 an G 2 ; G 1. W call G 0 th intrsction of th covr. If both G 1 an G 2 ar isomtric, thn w call G 1, G 2 an isomtric covr, an if both G 1 an G 2 ar convx, thn w call G 1, G 2 a convx covr. Lt G 1, G 2 b a covr of a connct graph G with G 0 = G 1 \ G 2. Clarly, G 0 is convx if an only if both G 1 an G 2 ar convx. Not that G 1 an G 2 may both b isomtric, but not G 0 which mayvn b isconnct. Lt G 0 b a connct graph, an lt G 0 1, G 0 2 b a covr of G 0 with G 0 0 = G 0 1 \ G0 2. Th xpansion of G 0 with rspct to G 0 1, G 0 2 is th graph G construct as follows. Lt G i b an isomorphic copy of G 0 i,fori =1 2, an, for any vrtx u0 in G 0 0, lt u i b th corrsponing vrtx in G i,fori =1 2. Thn G is obtain from th isjoint union G 1 [ G 2, whr for ach u 0 in G 0 0 th vrtics u 1 an u 2 ar join by an g s Fig. 2. For xampl, if G 0 = G 0 1 = G 0 2 = Q n, thn G = Q n+1. If G 0 is a tr, an if G 0 1 = G 0 an G 0 2 consist of a singl vrtx u0 of G 0, thn G is th tr obtain from G 0 by aing a nw vrtx pning at u 0. Lt G 0i b th subgraph of G i in G corrsponing to th subgraph G 0 0 = G 0 1 \ G0 2 in G 0,fori =1 2, an lt F = F 12 b th st of gs btwn G 1 an G 2 in G, cf.fig. 2. Thn F is a matching, which inucs an isomorphism btwn G 01 an G 02 n by (u 1 )=u 2, for anygu 1 u 2 in F. Furthrmor, it follows that G 01 [G 02 togthr with F form a subgraph of G isomorphic to G 0 02K 2. Not that this notion of xpansion is call binary Cartsian xpansion in th much mor gnral stting of [21]. Th following lmma follows straightforwar from th nitions. Lmma 3.1 Lt G b th xpansion of a connct graph G 0 with rspct to th covr G 0 1 G0 2. Thn G 0 1 an G0 2 ar both isomtric in G0 if an only if G 1 an G 2 ar both convx in G. Th convrs notions to covr an xpansion ar, rspctivly, split an contraction. Lt G b a connct graph. Lt G 1 an G 2 b isjoint subgraphs of G, an lt F = F 12 b 5

G 0 '' $ $ '' $ ' G G 0 1 G 0 2 G 1 G 2 & & u 0 u %% & & 1 u %& 2 G 0 0 G 01 F 12 G02 $ % Figur 2: Graph G as th xpansion of G 0 with rspct to G 0 1 G0 2 th st of gs btwn G 1 an G 2, whr G 0i is th subgraph of G i consisting of th n vrtics of th gs in F, for i = 1 2. Thn G 1 G 2 is a split in G if G 1 an G 2 togthr with F form th whol of G, an F is a matching inucing an isomorphism along its gs btwn G 01 an G 02. Not that G 01 an G 02, togthr with F, form a graph isomorphic with G 01 2K 2. Th subgraphs G 1 an G 2 ar th sis of th split. In Fig. 2whav a graph G with split G 1 G 2 an a contraction G 0 with covr G 0 1 G0 2. Not that both sis of th split ar convx if an only if both sis ar isomtric. W call G 1, G 2 a convx split if both sis of th split ar convx. Now Lmma 3.1 can b altrnativly formulat as follows: if G is th xpansion of a connct graph G 0 with rspct to th covr G 0 1 G0 2, thn G 0 1, G 0 2 is an isomtric covr of G 0 if an only if G 1, G 2 is a convx split in G. Th nxt lmma follows immiatly from th nitions. Lmma 3.2 Lt G b a connct graph, an lt G 1 G 2 b a split in G. Thn G 1 G 2 is a convx split in G if an only if G 1 = hw uv i an G 2 = hw vu i for ach g uv with u in G 1 an v in G 2. Th contraction of G with rspct to th split G 1 G 2 is th graph G 0 obtain from G by contracting ach g of F to a singl vrtx. Clarly, if G is th xpansion of G 0 with rspct to th covr G 0 1, G 0 2 of G 0, thn, with th abov notation, G 1 G 2 is a split in G, an G 0 is th contraction of G with rspct to th split G 1 G 2, an vic vrsa. Nxt w stablish som basic facts about th bhaviour of subgraphs with rspct to xpansions or contractions. Lt G b th xpansion of a connct graph G 0 with rspct to th covr G 0 1, G 0 2 with G 0 0 = G 0 1 \ G0 2. Lt H 0 b an inuc subgraph of G 0 such that H 0 \ G 0 0 is nonmpty. Thn H 0 \G 0 1 H0 \G 0 2 is a covr of H 0. Lt H b th xpansion of H 0 with rspct to H 0 \G 0 1 H0 \G 0 2. Thn H is an inuc subgraph of G such that H \ G 1 H\ G 2 is a split in H an th union of H \ G 01 an H \ G 02 inucs a subgraph of G isomorphic to [H 0 \ G 0 0]2K 2. Convrsly, H 0 is th contraction of H with rspct to th split H \ G 1 H\ G 2. Clarly, H is connct 6

in G if an only if H 0 is connct in G 0. Th nxt lmma follows straightforwarly from th nitions. Lmma 3.3 Lt G 0 baconnct graph, lt G 0, 1 G0 banisomtric covr of 2 G0, an lt G b th xpansion of G 0 with rspct to this covr. Lt H 0 b a subgraph of G 0, an lt H b th xpansion in G with rspct to th covr H 0 \ G 0, H 0 1 \ G 0 2. Thn H is isomtric, rsp. convx, in G if an only if H 0 is isomtric, rsp. convx, in G 0. Lt G b a connct graph, an lt G 1, G 2 b a split in G whr F th st of gs btwn th sis of th split. Lt H b an inuc subgraph of G with both H \ G 1 an H \ G 2 nonmpty. Thn th subgraphs H 1 = H \ G 1 an H 2 = H \ G 2 form a split in H. Lt F H b th st of gs btwn th sis of this split in H, an lt H 0i not th ns of F H in H i, for i =1 2. Thn w hav H 0i H \ G 0i, but w o not ncssarily hav H 0i = H \G 0i,fori =1 2 cf. Fig. 3. Lt G 0 b th contraction of G with rspct to th split G 1, G 2. Thn th subgraph H 0 of G 0 prouc by this contraction is prcisly th contraction of H with rspct to th split H 1 = H \ G 1, H 2 = H \ G 2 of H. Not, howvr, that H is not ncssarily th xpansion of H 0 with rspct to H 0 \ G 0 1 an H 0 \ G 0 2. This is so bcaus H is th xpansion of H 0 with rspct to H 0 \ G 0, H 0 1 \ G 0 if an only if H 2 0i = H \ G 0i, for i =1 2. A simpl xampl is givn in Figur 3, whr H 00 is th corrsponing xpansion of H 0. ' & $ u u ' u u uu u u u %& $ G G 0 % ' & $ u ' u $ u uu u u & % G 1 G 2 G 0 1 G 0 2 H H 0 H 00 Figur 3: Expansions an contractions of subgraphs Again w sthath is connct in G if an only is H 0 is connct in G 0. 7

Lmma 3.4 Lt G b a connct graph, an lt G 1, G 2 baconvx split in G. Lt H b a convx subgraph of G with nonmpty H i = H \ G i, for i =1 2. If uv is an g with u in G 1 an v in G 2, thn u is in H if an only if v is in H. Proof. Not that, sinc H is convx, thr is an g xy in H with x in G 1 an y in G 2. Assum that u lis in H. By Lmma 3.2, w hav thathw vu i = hw yx i = G 2, so that v is on a gosic btwn u an y. By th convxity of H, w s that v is in H as wll. Similarly, if v lis in H, w uc that u lis in H too. 2 Th nxt corollary combins som of th abov lmmata an obsrvations. Corollary 3.5 Lt G b aconnct graph, lt G 1, G 2 baconvx split in G with contraction G 0, an lt H b a subgraph of G with th corrsponing contraction H 0. If H is convx in G thn H 0 is convx in G 0. Proof. Lt H b a convx subgraph of G, an assum that th subgraphs H \ G 01 an H \ G 02 ar nonmpty. It sucs to show that, for ach vrtx in H \ G 01, its nighbor in G 02 lis in H. Bcaus of th convxity ofh, thr xists an g xy with x in H \ G 1 an y in H \ G 2. Choos any vrtx u in H \ G 01, an lt v b its nighbor in G 02. Bcaus G 2 is convx, vry gosic btwn v an y lis in G 2. So v is on a gosic btwn u an y, whnc v is in H. Similarly, w uc that, if v is a vrtx in H \ G 2 with nighbor u in G 1, thn u is also in H. By th prvious obsrvations w infr that H 0 is also convx. 2 Thorm 3.6 Agraph is a smi-mian graph if an only if it can b obtain from th onvrtx graph K 1 by xpansions with rspct to isomtric covrs with connct intrsction. Proof. First assum that G is a smi-mian graph. W will show that G can b obtain from a smi-mian graph with fwr vrtics by an xpansion with rspct to an isomtric covr with a connct intrsction. Lt ab b any g of G, an lt G 1 = hw ab i, an G 2 = hw ba i, an G 01 = hu ab i, an G 02 = hu ba i. From Lmma 3.2, Thorm 2.2 an Corollary 3.5 it follows that G 1 G 2 is a convx split with connct G 01 an G 02. Lt G 0 b th contraction of G with rspct to this split. Thn G 0 1, G 0 2 is an isomtric covr of G 0 with nonmpty intrsction G 0 0, an G is th xpansion of G 0 with rspct to this covr. It sucs to show that G 0 is a smi-mian graph. Lt u 0 v 0 b any g of G 0, an lt uv b an g in G corrsponing to u 0 v 0, an lt H 1 = hw uv i an H 2 = hw vu i an H 01 = hu uv i an H 02 = hu vu i. Lt H 0 i b th contraction of H i, an lt H0i 0 b th contraction of H 0i, for i =1 2. Thn H 1, H 2 is a convx split in G, hnc, by Corollary 3.5, H 0 1 an H 0 2 ar convx in G 0. This implis that H 0 1 consists of all vrtics narr to u 0 than to v 0 in G 0, an that H 0 2 consists of all vrtics narr to v 0 than to u 0. Morovr, w hav in G 0 that hu uv i = H01, 0 which is connct, bcaus H 01 is. Finally, w may conclu that H1, 0 H 0 2 is a convx split in G 0,sothatthgsbtwn th sis of this split form a -transitiv -class containing th g u 0 v 0. From ths facts w conclu that G 0 is a smi-mian graph. 8

Convrsly, lt G b th xpansion of a smi-mian graph G 0 with rspct to an isomtric covr G 0 1 G0 2 of G 0 with connct intrsction G 0 0. Thn G 1 an G 2 ar convx in G, so that F 12 is -transitiv. Furthrmor, for any g ab in F 12, th subgraph hu ab i is connct, bcaus it is isomorphic to G 0 0. Lt uv b any g in G not in F 12, an lt u 0 v 0 b th g in G 0 corrsponing to uv. Lt H 0 = hw u 0 u 0 v 0i an H 0 = hw v 0 v 0 u 0i in G0. Thn, by Thorm (ii), H 0 an H 0 ar convx subgraphs of u 0 v 0 G0. Lt H u b th xpansion of H 0 0 u with rspct to th covr H 0 \ u 0 G0 1, H 0 \ u 0 G0 2, an, similarly, lth v b th xpansion of H 0 0 v with rspct to th covr H 0 \ v 0 G0 1, H 0 \ v 0 G0 2. Thn H u an H v ar isjoint subgraphs of G, th union of which contains all vrtics of G. By Lmma 3.3, both H u an H v ar convx in G. This implis that H u = hw uv i an H v = hw vu i. Morovr, H 0u = hu uv i an H 0v = hu vu i ar connct an match by th st of gs F uv btwn H u an H v. Hnc F uv is a -transitiv -class. From this w uc that G is a smi-mian graph as wll. 2 4 Mian graphs vrsus smi-mian graphs In this sction w charactriz mian graphs in trms of smi-mian graphs. For this purpos w stuy convxity concpts in bipartit graphs, an, in particular w introuc th notion of wakly 2-convx subgraphs. This notion also allows us to slightly strngthn on of th known charactrizations of mian graphs. Lt H b a subgraph of a graph G. Thn th bounary @H of H in G is th st of all gs xy of G with x 2 H an y =2 H. Th following rsult is from [13]. Lmma 4.1 (Convxity Lmma) An inuc connct subgraph H of a bipartit graph G is convx if an only if no g of @H is in rlation to an g in H. Lt H b an inuc connct subgraph of a graph G. Thn H is 2-convx if for any two vrtics u an v of H with G (u v) =2,vry common nighbor of u an v blongs to H. Th subgraph H is wakly 2-convx, if for any vrtics u an v of H with H (u v) =2 vry common nighbor of u an v blongs to H. Clarly, a 2-convx subgraph is a wakly 2-convx subgraph. Th convrs is not tru, consir for instanc th path on v vrtics as a subgraph of th 6-cycl C 6. A graph G =(V E) is call msh if it satiss th quarangl proprty: for any vrtics u v w z with (u v) =(u w) =(u z) ; 1 an (u v) = 2 an z a common nighbor of v an w, thr xists a common nighbor x of v an w with (u x) =(u v) ; 1. Lt G b a connct, bipartit graph, lt u b a vrtx of G an lt P = v 1! v 2!!v k b a path in G. W call th g v i v i+1 on P upwar with rspct to u if (u v i+1 )= (u v i )+1 an ownwar if (u v i+1 ) = (u v i ) ; 1. Th istanc of u to P is (u P ) = P k i=1 (u v i). Lmma 4.2 Lt G b a connct, bipartit, msh graph an lt H b a subgraph of G. Thn th following statmnts ar quivalnt: (i) H is a convx subgraph of G, (ii) H is a 2-convx subgraph of G, (iii) H is a wakly 2-convx subgraph of G. 9

Proof. Th implications (i) ) (ii) ) (iii) ar trivial. It rmains to prov that (iii) implis (i). Lt H b a wakly 2-convx subgraph of G an assum that H is not convx. Lt u v b vrtics in H such that I(u v) is not contain in H with (u v) = k as small as possibl. Not that k 3. By th minimality of k, thr is a u v-gosic Q of which all intrnal vrtics ar outsi H. Lt x b th nighbor of v on Q. Sinc H is connct, bing wakly 2-convx, thr xists a u v-path in H. Lt P b a path from v to u in H that minimizs th istanc from u to P. Suppos that P contains an upwar g with rspct to u. Thn thr xists a subpath p! q! r of P such that pq is upwar an qr is ownwar. Now, G bing msh, thr xists a common nighbor s of p an r with (u s) = (u p) ; 1. Sinc p! q! r lis in H, anh is wakly 2-convx, s also lis in H. But now th path P 0 obtain from P by rplacing q by s has smallr istanc to u, which crats a conict with th minimality ofp. So P contains only ownwar gs, that is, P is also a u v-gosic. Lt y b th nighbor of v on P. Thn, as G is msh, w n a common nighbor w of x an y with (u v) = (u y) ; 1 = (u v) ; 2. Sinc x is not in H, an sinc H is wakly 2-convx, w cannot b in H. Hnc I(u y) is not contain in H, which is impossibl by th minimality ofk = (u v). 2 Not that a graph is msh if vry tripl of vrtics has a mian. Hnc, by Lmma 4.2, w hav th following rsult. Corollary 4.3 Lt G b a connct, bipartit, graph in which vry tripl of vrtics has a mian. Thn a subgraph H of G is convx if an only if H is 2-convx. Corollary 4.3 is givn in [14] in a wakr form, stating that convx is quivalnt to isomtric an 2-convx. In [17] it is prov that a graph G is a mian graph if an only if G is a bipartit, msh graph with 2-convx intrvals. Thus, from Lmma 4.2 w also infr th following charactrization of mian graphs: Corollary 4.4 Lt G b aconnct graph. Thn G is amian graph if an only if G is a bipartit, msh graph with wakly 2-convx intrvals. Almost-mian graphs wr introuc in [13] as partial cubs for which th subgraph hu ab i is isomtric for any gab of G anitwas prov that a graph is a mian graph if an only if it is an almost-mian graph which contains no convx Q ; 3 as a subgraph. W ar now abl to strngthn this rsult as follows. Thorm 4.5 A graph G is a mian graph if an only if G is a smi-mian graph an contains no convx Q ; 3 as a subgraph. Proof. Clarly, a graph which contains a convx Q ; 3 cannot b a mian graph. Suppos G is a smi-mian graph which contains no convx Q ; 3. Thn G can b construct from th 1-vrtx graph by a sris of connct xpansions. If ths xpansions ar also convx, w obtain a mian graph. Thus, thr must b a rst xpansion stp which 10

las to a non-mian graph. Sinc all graphs construct up to this stp ar mian graphs, this xpansion cannot b convx by Mulr's convx xpansion thorm. To mak notation asir, lt us assum for th tim bing that th last xpansion stp is th only on not convx in hw uv i. As hw uv i is a mian graph, U uv cannot b 2-convx ithr by Corollary 4.3. Thus, thr is a squar consisting of th gs ab bc c a in hw uv i whr a b c ar in U uv, but not. Lt a 0 b 0 c 0 b th nighbors of a b c in U vu. Thn a b c a 0 b 0 c 0 form a convx Q ; 3 in G, which raily follows from th Convxity Lmma 4.1. If this is not th last xpansion stp, w not that all furthr connct xpansions a gs which blong to irnt -classs from all prvious ons, so this Q ; 3 rmains convx by Lmma 4.1. 2 5 Mian graphs vrsus partial cubs In this sction w will answr th qustion which aitional conitions a partial cub must satisfy in orr to b a mian graph. For this sak w rst stuy th intrrlation of partial cubs with smi-mian graphs. An vn graph is a graph whos vry vrtx has vn gr. Rcall that nots th symmtric sum of graphs, whr w consir graphs as sts of gs. Lt C b a cycl of a graph G. Thn a st of 4-cycls C = fc 1 C 2 ::: C p g in G is a tiling of H if H = C 1 C 2 C p : Call a graph G til if vry cycl of G has a tiling. This nition nabls us to formulat th following rsult about partial cubs an smi-mian graphs. Thorm 5.1 A graph is a smi-mian graph if an only if it is a til partial cub. Proof. Lt G b a til partial cub which is not a smi-mian graph. Sinc G is not a smi-mian graph, thr xists an g ab 2 E(G) such that hu ab i (an hu ba i) is not connct. Lt hu 1 i an hu 2 i b two irnt connct componnts of hu ab ab abi an lt hu 1 i ba an hu 2 i b th appropriat connct componnts of hu ba bai. Lt u 1 v 1 an u 2 v 2 b gs from F ab with u i 2 hu i i an v ab i 2 hu i i for i = 1 2. W may assum that u ba 1 an u 2 ar as clos as possibl unr th abov assumptions. Thn thr is an inuc (in fact vn isomtric) cycl C = u 1 Pu 2 v 2 Qv 1 u 1 of G such that th paths P an Q li in W ab n U ab an W ba n U ba, rspctivly. Obsrv rst that th lngth of C is at last 6. Sinc G is til, thr xists a st of cycls C = fc 1 ::: C p g (p 2) of G such that C = C 1 C p. Not that if th 4-cycl C i has an g in F ab, thn it has xactly two gs in F ab. Sinc u 1 v 1 an u 2 v 2 ar th only two gs of C which ar in F ab thr xists a subst C r = fc i1 ::: C ir g of C, whr vry C ij has two gs in F ab an F ab \E(C i1 C ir )=fu 1 v 1 u 2 v 2 g. But thn w obsrv that in C i1 C ir thr is an u 1 u 2 -path with all innr vrtics in hu ab i. This is a contraiction, for w assum that ths two vrtics ar in irnt connct componnts of hu ab i. Suppos now thatg is a smi-mian graph. Thn G is a partial cub. W n to show that G is til. Th proof is by inuction on th numbr of -classs of G. If thr is no 11

such class thn G is th on vrtx graph, which istil. Now, lt G b a smi-mian graph with at last on -class. Thn G can b obtain by a connct xpansion stp from a smi-mian graph G 0, which is til by th inuction hypothsis. Lt F ab b th -class obtain in th xpansion stp. Suppos that C is an arbitrary cycl of G. W will prov that C is til by inuction on je(c) \ F ab j. Lt C 0, W 0, an W 0 corspon in ab ba G0 to C, W ab,anw ba, rspctivly. Suppos rst that je(c) \ F ab j =0. Thn w may assum that C is a cycl of hw ab i. By th inuction hypothsis thr is a tiling C 0 = fc 0 1 ::: C0 pg of C 0 in G 0. Lt C 0 b b th st of 4-cycls of C 0 which hav avrtx in W 0 n W 0 an lt ba ab C0 a = C 0 nc 0. b Lt C b b th st of 4-cycls in hw ba i which naturally corspon to th 4-cycls of C 0. b Similarly, lt C a b th st of 4-cycls in hw ab i which corspon to th 4-cycls of Ca. 0 St C = C a [C b. Not that th graph H b = D2Cb D is an vn subgraph of U ba. Lt H a b th isomorphic copy inu ab of H b. Sinc C 0 = C 0 C0 1 p, it follows that H a = C D2Ca D: (1) Lt C ab b th st of 4-cycls D = cfc such that c 2 H a, f 2 H b, an fc 2 F ab. Sinc H a an H b ar vn graphs, by (1),w conclu that C a [C b [C ab is a tiling of C in G. Assum nxt that E(C) \ F ab = fu 1 v 1 u 2 v 2 g, whr u 1 u 2 2 U ab an v 1 v 2 2 U ba. Lt P u b an arbitrary path in hu ab i btwn u 1 an u 2. Dnot by P v th isomorphic copy of P u in hu ba i. Not that P u an P v always xist sinc hu ab i an hu ba i ar connct graphs as G is a smi-mian graph. Lt H = P u [ P v [fu 1 v 1 u 2 v 2 g. Thn H is a cycl an it is asy to s that thr xists a tiling C H such thatach 4-cycl of C H has two gs in F ab, on in P u an on in P v. Not that w can consir th vn graph C H as a st of pairwis g isjoint cycls (possibly mpty st)such that ach on is ithr in hw ab i or hw ba i. Thus, by th prvious cas, for ach such cycl D 2 C H thr xists a tiling C D. Lt C CH = D2CH C D : (2) Now it is asy to s that C CH is a tiling of C H: Finally, obsrv that C CH [C H is a tiling of C. Suppos now that E(C) \ F ab = fu 1 v 1 ::: u 2t v 2t g (t 2) an that vry cycl D of G with jf ab \ Dj < 2t is til. Th argumnt is basically th sam as th prvious on. W may assum that thr is a path P u btwn u 1 an u 2 in C n F ab. Lt P v b an arbitrary path in hu ba i btwn v 1 an v 2. Lt H = P u [ P v [fu 1 v 1 u 2 v 2 g. Sinc je(h) \ F ab j =2,it follows by th abov that thr xists a tiling C H for H. Similarly as abov consir th vn graph C H as a st of cycls such that ach on has at most 2(t ; 1) gs from F ab. Thus for ach cycl D 2 C H thr xists a tiling C D. Lt C CH b n as in (2). Thn, as abov whav thatc CH is a tiling of C H an hnc C CH [C H is a tiling of C. 2 Combining Thorm 5.1 with Thorm 4.5 w obtain th following charactrization of mian graphs as partial cubs. Corollary 5.2 A graph G is a mian graph if an only if G is a til partial cub which contains no convx Q ; 3. 12

6 An Eulr-typ formula for smi-mian graphs Lt G b a mian graph with n vrtics, m gs an k quivalnc classs of th rlation. It was prov in [16] that 2n ; m ; k 2, an that quality hols if an only if G is a cub-fr mian graph. Th purpos of this sction is to xtn this Eulr-typ rsult to smi-mian graphs. For this sak w n th following corollary to Thorm 3.6. Corollary 6.1 A graph is a (K 2 2C 2t )-fr smi-mian graph, t 2, if an only if it can b obtain from th on vrtx graph by a connct xpansion procur, in which vry xpansion stp is takn with rspct to a covr with a tr as intrsction. Proof. Lt G b a (K 2 2C 2t )-fr smi-mian graph which isaconnct xpansion of G 0 with rspct to G 0 1 an G 0 2. Thn G 0 0 = G 0 1 \ G0 2 contains no cycl bcaus G is (K 2 2C 2t )- fr. Morovr, as G is a smi-mian graph, G 0 0 is connct an hnc a tr. Inuction complts th argumnt. Convrsly, assum that G can b obtain from th on vrtx graph by a connct xpansion procur, in which vry xpansion stp is takn with rspct to a covr with a tr as intrsction. By th xpansion thorm G is a smi-mian graph an w show by inuction that G is (K 2 2C 2t )-fr. Lt G b obtain from G 0 with rspct to G 0 1 an G 0 2 by an xpansion stp. As G 0 is a (K 2 2C 2t )-fr graph, G can only contain an inuc K 2 2C 2t which intrscts both G 1 an G 2. Each 4-cycl K 2 2K 2 as a subgraph of K 2 2C 2t has two vrtics in G 1 an two ing 2. But thn w conclu that C 2t is a subgraph of G 1 an of G 2, thus G 0 1 \ G0 2 is not a tr, a contraiction. 2 W cannow stat th main rsult of this sction. Thorm 6.2 Lt G b a smi-mian graph with n vrtics, m gs an lt k b th numbr of quivalnc classs of. Thn 2n ; m ; k 2 : Morovr, 2n ; m ; k =2if an only if G contains no K 2 2C 2t with t 2 as a subgraph. Proof. W prov th inquality by inuction on th numbr of vrtics using Thorm 3.6. Th inquality rucs to 2 2ifG = K 1. So assum that G is th connct xpansion of th smi-mian graph G 0 with rspct to its isomtric subgraphs G 0 1, G 0 2 with G 0 0 = G 0 1 \G0 2. By inuction w hav 2n 0 ; m 0 ; k 0 2 for G 0, whr k 0, n 0, m 0 ar th corrsponing paramtrs of G 0. Lt t b th numbr of vrtics in G 0 0,sothatG 0 0, bing connct, has at last t ; 1 gs. Thn w hav n = n 0 + t an m m 0 +2t ; 1. Morovr, an xpansion stp yils on mor -quivalnc class an so w hav k = k 0 +1. Thus 2n ; m ; k 2(n 0 + t) ; (m 0 +2t ; 1) ; (k 0 +1) = 2n 0 ; m 0 ; k 0 2 : Clarly, w hav quality if an only if, all th xpansions hav bn takn with rspct to two isomtric subgraphs having a tr as intrsction. By Corollary 6.1, this is quivalnt with G bing a (K 2 2C 2t )-fr smi-mian graph. 2 13

Corollary 6.3 Lt G b a planar smi-mian graph with n vrtics, k quivalnc classs of th rlation an f facs in its planar mbing. Thn f n ; k. Morovr, f = n ; k if an only if G is (K 2 2C 2t )-fr. Proof. Combin th Eulr's formula n ; m + f = 2 with Thorm 6.2. 2 7 Two problms At th n of th papr w wish to stat th following two problms. Problm 7.1 Is vry rgular partial cub a Cartsian prouct of vn cycls an K 2 's? Problm 7.2 Is vry rgular smi-mian graph a hyprcub? Rcall that Mulr [20] sttl th lattr qustion armativly for mian graphs. Not also that th answr is ngativ for partial cubs (consir C 6 ). Finally, obsrv that by Thorm 5.1, th truth of th rst problm implis th truth of th scon on. Rfrncs [1] F. Aurnhammr an J. Hagaur, Computing quivalnc classs among th gs of a graph with applications, Discrt Math. 109 (1992), 3{12. [2] H.-J. Banlt, Rtracts of hyprcubs, J. Graph Thory 8 (1984), 501{510. [3] V. Chpoi, -Convxity an isomtric subgraphs of Hamming graphs, Cybrntics 1(1988), 6{9. [4] V. Chpoi, Sparation of two convx sts in convxity structurs, J. Gom. 50 (1994), 30{51. [5] F.R.K. Chung, R.L. Graham, an M.E. Saks, Dynamic sarch in graphs, in: Discrt Algorithms an Complxity (H. Wilf,.) (Acamic Prss, Nw York, 1987), 351{387. [6] V. Chpoi an S. Klavzar, Th Winr inx an th Szg inx of bnznoi systms in linar tim, J. Chm. Inf. Comput. Sci. 37 (1997), 752{755. [7] D. Djokovic, Distanc prsrving subgraphs of hyprcubs, J. Combin. Thory Sr. B 14 (1973), 263{267. [8] T. Fr, Prouct graph rprsntations, J. Graph Thory 16 (1992), 467{488. [9] R. L. Graham, Isomtric mbings of graphs, in: Slct Topics in Graph Thory III, (L.W. Bink an R.J. Wilson, s.) (Acamic Prss, Lonon, 1988), 133{150. [10] R. L. Graham an P. M. Winklr, On isomtric mbings of graphs, Trans. Amr. Math. Soc. 288 (1985), 527{536. 14

[11] R. Graham an H. Pollak, On th arssing problm for loop switching, Bll Systm Tch. J. 50 (1971), 2495{2519. [12] R. Graham an H. Pollak, On mbing graphs in squash cubs, in: Graph Thory an Applications, Lctur Nots in Math., Vol. 303 (Springr, Nw York, 1972) 99{110. [13] W. Imrich an S. Klavzar, A convxity lmma an xpansion procurs for bipartit graphs, Europan J. Combin., in prss. [14] P.K. Jha, G. Slutzki, Convx-xpansion algorithms for rcognizing an isomtric mbing of mian graphs, Ars Combin. 34 (1992), 75{92. [15] S. Klavzar, I. Gutman, an B. Mohar, Labling of bnznoi systms which rcts th vrtx{istanc rlations, J. Chm. Inf. Comput. Sci. 35 (1995), 590{593. [16] S. Klavzar, H.M. Mulr an R. Skrkovski, An Eulr-typ formula for mian graphs, Discrt Math. 187 (1998), 255{258 [17] S. Klavzar an H.M. Mulr, Mian graphs: charactrizations, location thory an rlat structurs, J. Combin. Math. Combin. Comp., in prss. [18] H.M. Mulr, Th structur of mian graphs, Discrt Math. 24 (1978), 197{204. [19] H.M. Mulr, Th Intrval Function of a Graph (Mathmatical Cntr Tracts 132, Mathmatisch Cntrum, Amstram, 1980). [20] H.M. Mulr, n-cubs an mian graphs, J. Graph Thory 4 (1980), 107{110. [21] H.M. Mulr, Th xpansion procur for graphs (in: R. Bonik (.), Contmporary Mthos in Graph Thory, Wissnschaftsvrlag, Mannhim, 1990) 459{477. [22] E. Wilkit, Isomtric mbings in Hamming graphs, J. Combin. Thory Sr. B 50 (1990), 179{197. [23] E. Wilkit, Th rtracts of Hamming graphs, Discrt Math. 102 (1992), 197{218. [24] P. Winklr, Isomtric mbings in proucts of complt graphs, Discrt Appl. Math. 7 (1984), 221{225. 15