itIt

lt

I

On binary matroids with a K3,3-minor

S.R. Kingan

ABsaR^cT. W€ p|@ rhtt mry bi.lry 3-conn .red Datroid wirh d M{r(sr)-hinor mut it@ b.6 tn M(r(!\.). o! M'(I(.\.)-Biior, rh€ only qoption.b.inr M(K!!), lhs hiq!€ llltft€r 8ro for r€sure m.r.oir|!, &d s10€tmo,!,nrk-6, d€lf-durl E3!rcid cdlod M(E!).

1. Introduction

In 1930, Kuratowski [5] daracterized plarar graphs by proving ihar s graph isplaDar if and only if it has no subgraph homeomorphic with K6 or l(i,s. Wag-ner I10l proved that a graph is planar if and onty if ir ha.s no minor ircmor-phic to K6 or K3p. D.W. Hall [2] prcved that every simple g-connected gr&phwith a,L6-minor must also have a,f3,s-minor, the only exception b€ing K, it-s€U. Seymour [9, ?.5] noied that the above result abo hol& for reEular ma-troide. In [3] the audor sbowed rhar Hall's r6ult can be generatized ro binarymatroi& a! follows: Er€ry 3-connect€d bilary rnahoid with aD M(Kd)-minor musialso_have An M(Ks,s)- or M'(K3,s)-minor, rbe oDly exc€ptions beihg Ir(Kr), Ihigbly symmetric, l2-€l€ment, rank-6, binary mahoid called fr2, and any single-element contraction of 4r. This result implis that the exclud€d,Einor cla.$€st x(M (Ks3), M'(Ks,s\, M (K6), M'(K6),U2.41 bd C X(M (h,x), M. (&,s),Ur, . \are almost the sarne, wbere €l(M\M2,..,,M*) deDot6 th€ clats of matroidghaving nole of the matroids Mt, M2,, . . , ML as minors. Not€ that excluding Ur,.Suarante€s that these matroids are binary.

In this paper we study rhe excluded-minor class tx(M(fu),M.(fu),M (K'\e), M'(Kt\e),Ur,|). One reason for studyins this clajs b a! folbwj: Iris well kno.r'n thar th€ three nonisomorphic binary 3-connected single,elementextensions of the 4-wb€r-l M(Wa) are M(K5\e), M.(K3,s), and a nonregular 9-el€menr marroid Ce. In [0] Oxtey determined th€ 3-com6cted membe's of theclals t,l ( Pr, Ps' , Ur,. ). Therefore it would be int€r€ting to know the 3-connectedrn€mbers of tz(M(l(3,x), M'(K3i, M (&\e),M'(K6\e),U2,). We plo\€ thatthe abol€ class ar'd. tx(MlKs\e),M'((s\e),Ur,d are ajrnost rhe sam€. Themain th€oreI| of thi! paper staies ihat every 3,connecied binary matroid with

ts,4t u.tia.rt.t S@tet Clarf6r'on. Prim&y 0s835, o6c?s

o 'ee6 ̂ Dqkd M{hcrebd sdrliy

S. R- (INGAN

an M(f3,3)-minor Bwt also bave aD M(lfi\e)- or M'((r\e)-minor, the only ex-ceptions being M(f3,s), Rro, and M(Es). Binary matrix representations for Rroand M(Er) a.re shown below.

1234 5678910 r2 345678910

M \E"The matroid Rro is s€lf-dual and it b the unique splitter for regular rnstroids lgl.

The l ioear l ransformarion (rr , rr , ts, cr, ra)r --+ (rr , rr , er, rr , ro)r mimthe matrix r€prBenting M'(Ei) io the matrix representing M(tr); bence M(t5) isself-dual. It b non-regular since M(E )\5/4,8 e F?. Chula Jayar€rdan€, a studeDrof N€il Rob€rEon at Ohio State Univ€rgity, independently showed thai tbe mstroidM(86) ir a spliiier for the claaB of binary matroids wiih neiiher M(..(6\e)- norM'(l(s\e)-ninors (s€€ Lernha 2.3).

The matroid t€ininolory iD g€neral follows Oxl€y 14. A matroid lV is a minorof a matroid M if N ! M\X/Y for some diljoint subsets X and Y of E(M).A circuit with * elemenh b called a t-citr: ir. The clclc rndrroid of a graph CiB dehoted by M(G). The wctor matroid of a 'x,atrix I is denoted by M(,4). Amatroid b Dinory if it can be repr€sent€d by a matrix over df(2). A belic tool inthis paper i3 the well-known fact that binary matroids ars uniqu€ly r€presentable;thsi is, if ,4 and.4r are r x n matrices over cF(2) such that th€ map which,for all t € {1,2,...,11}, ta}€r th€ i th column of ,4 io the i th column of.4' isan isomorphism hom M(,4) to M(,4'), then,4'caD be transformed into,4 by asequence of operationr each of which consists of interchanging two roc,s or addingone ro$' to another. W€ shall assume familiariiy with the pivoiiDg op€raiion. Inoder to maintain a repr$€ntation in ihe standard forn [I.lr], ev€ry pivot will befollowed by the appropriate column int€rchan8e. See, for exarnpl€, [?, p. 209i. If .4is an r x 'l matrix rrith colurlo llbeb 1,2,...,fl, aEd t ir an r x I column v€.ior,then denote by ,4ut ihe r x (n + l) rnatdx A with ihe column i a6x€d at the €nd.Lab€l the colum$ of AU i aB 1,2,.. .,n,r. A m&troid M is s-connected if it i3connect€d and E(M) cannot be partiiioned irto subsetE X and y, earh having atleast t*,o clements, such that r(X) + r(y) - r(M) = l. Let C be a clas8 of rnatroidsthat is clded under minors and und€r isomorphisms. A member N ofd is called arplirter for C if .o 3-connected member of C har a pmper minor isomoryhic to ,ry.

2. Thc main theorem

The following theorem is the main result of this paper. Computaiions irvolvingsingle€lemeni extensione may be found in the Appendix.

THEoREM 2.1. Svppose M is o3 cor.rte.ted binory rnatroid uith on M(Ktll'minor. Then eitler M hos an M\K6\e)- or M'(Ks\e\-minor or M b i'onorphtcto MlKn), Rn, or M(Ea).

LDMMA 2.2. The matroitu Rrc and M(Eol ha:l,e neither M \Ks\")' norM'(&\e\- ' , inoB.

( ^

I 0 0 I l \r r 0 l 0 l

alll;l0 0 l l 0 /

l r 0 0 l \ /l l l 0 0 l I011 l0 l I k0 0 l r l l I1 0 0 I l / \

Rro

- ;

2 3 4 5 6 7 8 I l 0 5 2 3 4 6 7 8 I r 01100 rt l l 0 l0 l l l I0011r

ON BINARY MATROIDS \MITE A K3,3-MINOR

(r(i,3,ecr1)\t

5, 2,3, 4,6, ?,8, I, l0 of the matrix repres€ntingrhe ej'Ees a, b, c, d, L g, h, i, e of the $aph K6 in Figuree M(n5V). So if Mr = (ri,3, ertr)', then Mr has

FtcuRr 1. Ks.

pnoor. Since fro and M{86)are self-dual, it is suffcient to show rhat neirherof (hem h,l an M,i.Kr\e)-ninor. Every single-elemenr deletion of Rro i.s isomor-phic to M ( K. . |. Therefore RIo har no M'(K6\e)'minor' Th€ matroid M (E6 ) ha.sn i n . + a . . i , i * r r 2 . 3 , 6 1 . t 2 , 3 . 4 . 7 ) , 0 , 4 , 6 , 7 ] . { 3 . 4 , 5 . 8 } , { 2 . 5 , 7 . 8 } . { 4 , 5 , 6 , 9 } ,{1,5,7,9}. t t .2,8,9}, and {3'6.8'9} Each element other than l0 appeers in ex-acclv foui ,t.circuits. and l0 appears in none of th€ 4circuics Suppos€, if possible,

MtI .6t \e = M.{r{ . \?) forsorne element Pin M(Ei). Since iheci tor i tsof M(&)\eare rhe circuits ;f t (E5) not conraining €, M(E6)\e will have €itber ffve or nine4_circuits. This is a conrradiciion since M'(K6\e) has ibree rlcircuits. Therefore,M(E ) hal no lr''((r\e)'rninor' o

pRooF oF THEoREM 2.t. Suppose M isa3-conDecred binary rnarroid wich anMt& -minor. Sevmour's Splitter Theorem J7, l1.2.ll implies chat ch€re i! a chainMo, M,. . . , v. of 3-onneaed_ -atroids such that Mo ! M(Ks,sJ, M^ = M,and for al l i € {0.1.. . . .n - l } , Mirr i3 a si t rgle-element €xtension or co€xrensionof Mt. U M = MIki, then there is aothing to prove. Therefore, a$ume thisdoes not occur. Then. l/o : M(Krr) and Mr is an extenoion or coextension ofM{/(rrt. Suooose Mr is a coextensioD of M(K3.s). Note that the coextensions of.Vtrs 3t *e ii'" auas

"f rhe €xtensions of M'(K3.3) The matroid M'(Kr,s) haj

pr€cisely obe binary 3-contrected sinSleelement excensior denored by (Ki,3,€.ffl).

Th€ maLri@s r€Dresenting (Ki,r' extl) and (K;,3' ettl)\ I are shown belov'

I

:

l 0 0 l \ /110 r l lI l l l l l r1l I I 0 / \

II0I

( ,

Observ€ that columns(llj,3,eetl)\l corre5pend tol. Therefore ((j,3, errl)\l

366 S. R, KINGA}I

an M'(K5\e)-nirior, and conoequeDtly so does M. Next, supp6€ Mt is all exten-sion of M (K3 3 ) . The matroid M (ff3,3 ) bar precisely four non-isomorphic binarv3-connected singl+elemeni €r(ensions denoted by (/(s,3.e-rti) for i € {1.2,3.4i.Matrix represeDtations for th$e matroid3 are shown below.

L 2 3 4 56 7 t2 3 4 5l 0l l

0 l00,ettl(,(',g

( -

8 9 1 00 l r \ /

il;llI r o l l

) l t 0 / \

6III00

789r00 0 I r \l 0 l 0 ll r l l ll l l 0 l0 | | 0J

678910

l l 0 0 I l \l l I 0 I 0 |11 l I I I Il 0 1 I I 0 |l 0 0 l I l /rt4)

8 1 0 l l

3;;'\i ; ?l

t 2 3 45

I6

12 3 4 5

16

(&,s, ett2l

(Ks,s,e2

6 7 8 9 1 0I 0 0 r l \ /I r o r r l ll l l l 0 l l0 l l l l l l0 0 I 1 0 / \, ert\)(Ks,s

6 7 8 1 0 1 l rl 0 0 l 1 \ /11 0 0 0 l ll r I 1 0 l l0 l 1 0 1 / \

l 01 ll l0 0

( .

Ob!€rr€ that (&3.e{ll) + M(Ki,3). rhe cycte matroid of rb€ graph K3,3 wirhan addidonal edge (see I p. ll0]) and (.K3,3, €rr2) = M(Ed. Tbe linear iransfor_nation (tr,,r, rs, q, r)T --+ @t, t2 + q, 24 + zt, r., .5)" maps rh€ matrixrepres€trting (Ke,s,€st4) to rhe marrix repr€senring Rro; hence (Ki,3, d:t4) g Rro.tanma- 2,2 inplies iha! nro aDd M(Er) have no M(K6\e), or Mr1ro1"j_mino..Each ofthe marroids (Ks,r,er )/a and (Kr.3,ar3)/3 is isonorphi; ro itr{K"\e).lf Mr i! isomorphic ro eirher (K3,3, erfl) or (Kj,r, e"r3), iheD M, t ar * Virit.j_minor, and therefore so doe3 M. So assume Mr g R,o or M(Er). The next lernmacompl€tB the proof of the th€oren.

LEMMA2.3. The motmidt ho ond M(E !ft,ptitters lor the ctars ol 'an(lr,'natmidr uith ne et M(K6\e)- not M.(K6\e)-r'i.inors.

PRooF. The matroid &o has two nonisomorphic binary 3-connected ,ingle-elem€nt extensions denot€d by (Bro,er ) and (Rro, err2). It b shown in the ;ppendix thai (Rro,€.rtl) has a minor isomorphic ro M(Kj,3) and (Rro, ert2) hasa minor isomorphic to (Kr!,e.,t3), both of which have an M(K6\e)-minor. Thematroid M(ts) lus sev€n nonisomoFhic binary 3 cormected sinSle-eiement exten_sioDs deEoted by (A5,e.r i ) for i € (1,2,. . . ,7). Each of rhe mairoids (Es,e{ land (4, ert4) hae a minor isomorphic to M(K1,3). Each ofthe ruarroi& (E6,err3laitd {E , err6)_has- a minor isomorphic to (I(3,s,*tS). Il.jt (Es,e"t2), @;ert');,an_d- ( EE, ert7) b€ rhe marroids obtained by adjoioin8 cotumns (t0010), (j00Ol), and(l0l0l). respe^ctively, ro E6. The matrices representing rbe matroi& lE6, €rt2ilS\9.{!s,eris)/a\9, and (Es,er17)/4\9 are shown betowl

| 2 34 2 3 56 7

ON 8INARY MATROIDS wlTH A K3,3-MINOR

(86, ert2) l5\s\ 2 35

(&, ezts) /4\s6III0

( "

7810 1 r0 0 I l \l 0 0 0 lr l r 1 l0 l 0 l //4\e(86, * t7)

Observ€ that columN 1,2,3,4,6,7,6,10,11 of the mat x reprsenting(E6,ert2) l5\9 corresponds to the ej,ges l,j,a,e,c,d,h,6,t of the graph Ki

-in

Figure L Thereforc (Es,ert2)15\9 s M(&\€). Sirnilarly, each of ;himat;ids(E6,at5)/4\0 and (Ds,617)/4\9 is borDorphic ro M(K6\e). Iherefore. esch bi_nary 3-connect.d extcnsio'r of Rr0 and M(E6) ha, a minor isomorDbic ro M{K"\e).Findly. since Rro and M(Ej) are self-dual, each of their binary i-co".ertel ;;_tensions has a minor isomorphic to M'(L6\e). - tr

For Sraphs tbe marn rheorem implies ihar every simple &connect€d AraDb with1.rsq.ryno1mu11 atso have a ,(6\e- or ((r\e).- minor, the only excepiion beingIls,r itsell The follorving corollary is a direct consequance of the main iheorem.

CoRoLL^Ry 2.4. Th. 3..onn.cted binary matfttds n eXlMtKE\et.M'( i (6\e) ,ur not n tx lMlKs\e), M.(K6\e), MlKs3), M'(Kx.r j l of t p; .cbetu M(Ks.x), M'\Kn), Rrc, and M(E). t r

Appendix

Tho matroids M(.K3,3) aJld M'(K3,r) are reprelented by the marrices X andX', respectivelx shown below:

I . . . 5 6 7 8 I

" (' ii;ii)" (' "'iiiiirtAssume rhat the mlumn r = (zr,rr,4,xa,4)r b adjoined ro rhe matrix x roobtain a 3-connected binary matroid. For €onveni€nce we will droD the trans_pose. There are 22 choicer for r. l fr€{ut000).(0lt0O),(00tr0),(rtt lO),(OOOID.(0r r lr)), rhen M I x u rt = lKl l .er ). rf r € {(10100), (r0010), tororoi, irooori.{01001), (00101 ), ( l l l0l), ( l l0l l ) . { l 0l l l ) }, rhen M (X r r) = lK s t, ed2). l f t i( ( l l 0 l 0 ) , { l 0 l l 0 ) , ( l r 00 r ) , (0 01 ) , (100 ) , (010 ) } , rhenM( i l J ; i : rK ; " .e r r3 ) .If i = ( l0l0l), rhen M (X u t) e (r{s,3, e'r4). Observe that, (K3,3,e1riy ha.r three3-circuits and four 3-cocircuits, (K3,3,€rt2) has rwo 3,circuits and two 3-cocircuits.(K3.j,e"t3) has ?€ro 3-circuits and (wo 3-cocircuirs, aDd (Ks.3,ert4) haj zrro 3circuits and zero 3,cocirorits. Hence these extensions are nonisomomhic. To verifvthai cobmfi in each class give rbe Lo isorEorphic extensions of MiK3,31 obs.rvethat each of the following lin€ar rransformations on X inducer an automomhismon MlK3'3)l

a : ( q , x 2 , 4 , x t , x s ) T , - - - + l x t , q + x t , r t + r t , 4 + q , E 2 + r r ) 7 ,f : (sr, xz, xs, xe, zs)r t--+ (q + 22, ?2, ai + ar,

", +

"r, ," + ,rir,

368 S. R. KINCAN

) : ( r r ' " ? ,

1 3 , 3 { , r r ) r - r ( t r , t r + r r a t { , ? r ' t r r r 1 6 , 1 2 + r 3 ? r i . ' . ) rNow observe rhat d(ol l l l ) r = (1r000)r, o(11000)r = ( l to)r , ,0 io)r" ' -(01l00Jt, o(01100)r = (0001t)r , and d{00011)r = (00110)7. Therefore, adjo' in ingany one of these columns to X gives a matroid bomorphic to (K:,s,ectl). Theoth€r cases can be done similarly. lt can be checked that M'((3,3) ha! preciselyooe binary 3-conEected single-element extebsion.

Next, the mairoids Rro and M(ts) are reprerented by the matric€s y ard Z,respectiv€ly, showD below:

1 . . . 5 6789 r0I 0 0 1 1 \l l 0 l 0 lr l r r r l .0 l r I 0 l0 0 l I 0 /

678910r 0 0 r r \ /r r 0 r 0l I| | | | r I ,z= |0 l l l 0 l I0 0 l I l / \

"= (h

For conv€ni€nce, ftro is represented a.9 a single-elem€nt ext€nsion of M(K3.3).Assume that the colurnn t = (q,az,xs,xt,xa)T is adjoined to th€ marrix yro obtain a 3-connected binary matroid. There are 2l choices for i. If t €{( l l000), (10r00), (01100), (10010), (01010), (00110), (1 10), (r0001), (01001),(00101), ( l l t0 l) , (0001r), ( l l01l) , (10111), (01111)), then M(y u r) g (Rro, er ' l ) .I f t € { ( l l 0 l 0 ) , ( 1 0 1 r 0 ) , ( r r 0 0 r ) , ( 0 1 1 0 1 ) , ( r 0 0 1 1 ) , ( 0 1 0 1 1 ) } , t h e n M ( y u r ) =(Rro,ert2). There extensions ar€ nonisomorphic since (Rro,eatl) has three 3-circuits and zero 3-cocircuits and (rqlo,err2) hae ?€ro 3-circuits and ?,€ro s-cocircuits.Obs€.r€ tbat each of the follo$,in8 linear transformatiom on y induces an auto-morphism on Bro:

o 1 G r , a r , t i , r r , o r ) r r " + ( r r , q + r 3 , . r + ? 2 + ? 6 , x r * t : g , a r ) r ,0 : (r t , q, zs, zt , as)r , --+ (zr, q + a2t r t + ' ,11 x1t 4, xy +4\7.

These aucomorphisrru can be used to verify thst columns in €ach class give ris€to isomorphic €xtcnsions. Notice tbat (Rro,ertl) i! obiained by adjoining col-umn (11000) io f. Therefor€, (Rro,errl) ha! a minor isomorphic io (K33,ertl).Snnilarly, (Rlo,err2) is obtained by adjoining column (ll0l0) ro y. Therefore,(Rro,eir2) has a rninor isomorphic to (l(3,3,ert3).

Assume thai th€ colunn t = (q,x2,4,xa,x6Jr is adjoined to ihe mairixZ to obtain a 3-connected binary matroid. There are 2l choices for t. If t €{( 11000), (0r 100)}, tben M (Z u i ) = (86, e. tr) . f f r € {(r0010), (01010), (r101 t) ,( t01tI)) , r t rcn M(z u i ) = (Ei,est2). I f t € {(11010), (10110), ( l00t l ) , (0l0l l ) } ,.het! M(z u i ) = (E6,ert3). I f t € {(001r0),(r11r0),(00011),(0r lrr)} , rhenMlzu.) = (E5,e,t4). l f i € {(10001), (01001), (0010r), (rrr0r)} , rh€n M(zut) = (E5,e' t5). I r t € {( l l00l) ,(0r l0r)r ,rhen M(zui) = (Dd,€116). I f r =(r0l0l), then MlZUil = (E6,eEt7\. Obsewe that, (Er,ertl) has five 3-circuitsand two 3-cocircuits, (Ds,ert2) has four 3-circlriis and one 3-cocircuits, (Es,€ct3)has three 3-circuits and one 3-cocircuit, (Er, eaf4) has five 3-circuits and one 3-cocircuii, (t6,err5) has foui 3-circuits and zero s-cocircuits, (E:,ect6) has tro3-circuit3 and zero 3-cocircuits, and (E6,est7) bas three 3-circuits and zero 3,cocircuits. Hence these extensions are nonisomorphic, Observe that each of ihefollocing linear transformations on Z induc$ an automorphism oD M(95):

a : (zr, xz, rz, rt, xs\r ,---t lzz + xs, xt + l 3, x3, t4, e,s\ri l : ( t1, 7t , ," , xa, 4)" ' ' - t (4 + za, E2 + tE1, E.r + a'1, r ' t , ts + q)r

xs' xs)ru0)" =djoitring1). The)recisely

ON BINARY MATROIDS wlTH A X3J-MINOR 369

These automorphisrff can be wed to verify that colurns in e3ch class giv€ ds€ tohomorphic extensioDs,

AcEnowledgmeat: The author thaDk! Joe Bonin, Jam€r Oxley, and th€ refer€o formany helpfit cornments aDd suggestions.

R€ferences

B. Bolob&, dSrlrnol CBpl ?t@rr', A.ademic P!6, Ladon (r97E).D.w. ntll, A ^ot o^ tn^ili!. ,*.@ csracr! Bull. Ab6. M.rh. S@. t0 (1943), 935-937.S.R. Kinss, ,4 ,.ndlir.tion ol q rmph nult ot D.W, n.lr, Di!.!.t€ M.th.! t! .!I'N..J.P.S. KuDt! 8&ludin, aifr .rcL g@n.t'L. of tic Kn6tou*i .ftrit ton hinqt g@^.tri.,,Prcc. rrndon M.th. Soc. (3) 66 (rgE?),209-22.K, Kurrto$ki, 5!. ,. ptubun. dq co/th.t s6!.L, .n lopoiogt., !\d. M*h. r5 (19i0),z -243,

[6] J.G. Oaloy, "n.

oirrrr naafri.k vith no l-aidl ^inq, 'riin.. An€r. M!rh. Soc. l6a (19E7),6176.

[4 J.C. Oxl6y, (Martuid Ti@,y', Ortojd UDvdity P@, N€* YolL (re92).{81 N. R.bdt&n, P.D, S.yno!., ed.dlinns Ksntob*ri,, Ii.oE 4 consr6u! Num.r&tiuE

a6 {19E1), r29n38.[0] P.D. soyhorr, Daonrorition ol ds!!'/6. n6.fti.t , J. Codbh. Tleory S€r. a ,8 (t980), i05-

360,l10lI{. W.s!€!, Uh.r.i^. Eig.n,da& d.r.o.td rorptt , Ms!h. Ann. 114 (1937), 6?G690.

DEr^Rr E r oF M ̂ rB.!^arc^L ScrrNcrs, O^*L^I{D UN|vrfi3|"y, RocfissrEi, MI4E3O9C!ft^r .dtlfr.tt D€pNnhMt of Msthorrlo, W6ibltr.r€r ColleSe of S.tr L.ko Ctry, S.lr

Lelo ci!y, uT E t105&m6al od.hr..t t-lt!t.!arc.lc,.du

r0

lrlI2ll3llrl

l \0l1 l0 loJ

(Kr,a).'trix Ylf t €

,ertl\.

tlee 3-Lcuit3.auto-

!T

,r nseI col-ert l ) .

ratnxf i er0ll),) l l ) ) ,then'(z u'curtsettS)

:o 3-f tbe