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On matroidal families
!)iscr~ie MlillnHillllk
NOTE ON
MATIIOil)AL
IFAMII,IIES
M. I.(;I1.F,A Ihllt'fr~il~. Illm, fie llrlllvlb,s t' '~rh" #o~IM 2~1 ( fretful/ p!f,illt I/ I jl, Ilrudeldrd ,h* ilhm'pJl, ~ Jfi30 lJrl:xell~L Iil, IRftlll¢,
Ite'elved 2h FIdum!l) 11479 flllulle~.l~e t viral Im~ delhlvd i ~, ~, l I a m*t,r,fldal fmullv ,,r l!l,tphs rind ha< pmu d Ille e dq,.nu, i1{ t11111 ii1]nilltlfll btillilh!% called I,,, I!, i ~ ;Ira! ! , Ille! qL'l Ill lU'I}Ium~ [111 hinh/'ae III Itas qhl%{ll lhnl h,t eteiv ft. hllegeL il " 2, tllele If ft iiHllrlllllp; lmuilv ~f, fJ ,i Al~, I i Jl,I I¢ k lhe p0tl.u~e I,f thu ple%>lfl Illile ill hlllhl Zl ~el uf Illffltlllllffl frmfi,ie~ iiiillflltllrq~ rtll il,e M,, t, mid ith Illtilllh if ittlef ithtlufld*il lalfillle~
1, I~i~ltilitliilll
I;., ~, ~ti
tllllil:Oli~ ~; I~ it I~ilil lid, lit t,']l~ f~ if !~ Ft tlt!llt~ ~c,I ~ilid ,'b I~ it ~ilHl%:llull i,f "41h~lJh Ill I! h~ilt~d itltiTllfelld~lllf ~01tli ~t'fh th~ll fro, tl~l: It't ~if~ Uilftdlt~d z (tll tJt= Ji Ibt
l[ ,~Lf~
itlitl 7 ~: ~, Ihtsll i<~ #i
~lid, Ilfill I VUffll~
as.
A ~/lh~el ttf f7 llllf h~htitgillg In a ~. J~ ~1t~1 df~li~llfl#lll A s~ibs.vl ill i7 whicll i~ gll~lll~li0elll 1411~ ItlillJt11~II with !llis propl.~rty is called .~lig,ll. A rnillroJd M ctta ~lso he defilled ~s ii pair rE, 71 where ,<#' i~ a c(dleclicm Sllhae!s of E (cHImJ stiglllS) sttch lhat (a') [h'), It') are satisfied:
~i Ill
{a') ISl---1, w~,~; (h') If S iar, d S:~ are dPtinct rqerlbers c~f 7, then S i ¢ $2; (2') If S I and Sa are distinct mcmbe not contain any loops or multiple edges but the results are va'id for hypergraphs [ 3 4]. IU]
1,14
M totea
~;, The ma~rotfl MtG, ~, b) I,el (; t~e a si;t~t~le graph, a be an inleger (t~ - I), b be an integer (()'~1)~2.), trod +~' be file edge sel of G, The function [ :,~l'd'i-~ N, defined hy
i'~ nen decreasing, stibmodaiar [3] and thus [9, Chap;er gl it is possible m obtain a matroid on Ihe edge set '~' of G, by taking us in:Icpendent sets the sets X ( X c ~) such that, v g c .xj, f ( Y I ~ [ Y l, This matroid is called M(G, a, h),
3° Charac~erizat~on of the ~ligms of M(G, a, b) I,et ~ be contained in ~, the edge set c,f G. Theorem 1. 9, i~ a stigm of M(G, a, b) if and only if (il a ]Us~:, Sb- b = ]9,1-1, and (ii) 9, is minimal non empty with property (i). It is enough to prove that, if 3° satisfies conditions (i) and OiL it is dependent (part I) :rod that, if 9, is a stigm, it satisfies condition (i) (part 2). Part I. Condition (1) implies ([(9,)<]9,1), which implies 9' is not independent in M(G, a, b I. Purl 2. Lcl ff be a stigm (if .a, is strictly contained in ,~, ~ is independent and f(.~)~> [.r~[; it is thus aecessary to have f(9,)<19,]), we have thus:
aI~UlS
b~[9,]-!,
(1)
:if [9,1 - l, as f(9,)= a [Us~:.~Sl-b>~O, (1) implies
and (i) is proved. If 19'[> 1, let 9,' be contained in 9,, with 19,'1= [9,[-1, ~ ' is independent, and thus f(9,') ~ 19,'i, and alsUpSl-b~>lg, '-1,
(2)
(I) and (2i imply
19ol =t~,.
I
U s =,~'>.i U ,1=t,>1:/'1~1,
tin I lhus (i) is proved,
4, ConnectMty M th:" stigma, a¢ M(G, 0, h) Theocem 2, bor every graph G, ]'or e~ery integer , (a-~ 1), for ,.cry nleger b (2a ~ b ~ (I), and for every sliqm 9Oof M(G, 0, b), tile partial graph r?, G ]orme'l
by the edges helanging to cp is connected, We shall prove this theorem by contradiction: let us suppose there art~ st (k ~ connected cempoeents in the partial graph generated by the edges hehmging the stigm .~. Let V, be the num~er of vertices al~d E, be the numher of edges the ith con ~ected component. A-, 9 o L, a stigm, every connected compon~!D[ independent in the matroid M ( O , a, b) and
aE-b~E,,
21 a; If is
i = 1 , 2 . . . . . k.
Thus
o
V;-kb~E
e,,
and
o i U S[-b~lgol+
5 M(G, a, b) and the mat~oid'd families of Simoes-Pereira and A~dreav If we fix a and b(a and b integer a ~ > l , ant; 2a~>b~>0), the s igms cf all the matroids M(G. a, b) are the members of a matroidal family ~ee note this matroidal family MF(a, b); of course, the elements of MF(a, b) or. characterized by the conditions (i) and (it) of Theorem I. In [5, 6, 7], Simoes-Pen.ira has proved the existence of four matroidal families called Fo, F2, F=, and F 1, the set of polygons, and in [1] Andreae has shown that. for every n integer, n ~>2, there is a
IOf~
M. Lorea
m a t r o i d a l f a m i l y M,, c o n t a i n i n g t h e c o m p l e t e g r a p h wi~dl n v e r t i c e s . It is n o t difficult to) s e e t h a t :
Fc~ = M 2 - M F ( I , 2); F I = M 3 - M F ( I , 1); F , = M F ( I , 0)
M,.-MJ'(½n,
(Bicircular graphs);
½r~+l),
n i n t e g e r />2,
n ~, :n;
and
~,1~- M F t ~ ( n
1), 1),
n i n t e g e r />3,
n odd.
Reterences [I} "1 /,ndlcae, M~llroidal families of finite connected nonhomeomorphic graphs exist, J. Graph "fhe,~ry 2 (197S~ 149-153 [2] C I-crg,: Graplv; and Hypergraphs (North Nollan]. Amsterdam, 1973) [~] M I.orga. t.a ,,tructure de terrains problemes d'aafeetetion. Rev, Frangaise Automal, Inf'~rmat Recherche Ophrati,mnelle, Sdrie R,O. 10 (1976) S9 6~ [~] M lorda Matroldes ,ur les ensembles d'ar&tes d'hyp,:rgrapheb, Cahiers Centre l~tudes Recherche Ophr. 20 (1978; 127 t3:, [ ~] J M S . Simoes-Pereira, Cn malroids on edge sets of glaphs with connected subg*aphs as circuits II, Discrete Math. 12 (1975) 55-78. [hi J M S Sim~ms-Pcruira. Subgraphs m, circuits and bases of matroias, Discrete Math. 12 (1975) 79-'~g. [ 7 ~M S. Simoe~ Pereira, A comment on matroidal families, Colloques Internationaux C N.R2k, No. 2(dl Probl~me,~ comhinatoires et tlahorie des graphe~,, Paris-Orsay 1976, i978, 385-387 {8] W [ulte Inlroduction to the Theory of Malroids, Modern Analylic and Computational Melhods in Scicn~ ,rod Mathematics 37 (Elsevier, Amsterdam, 197D, [9] D Welsh. Ma[rtlld Theory (Academic Press, b n d o n New York-San Frmlcisco, 1976).
NOTE ON
MATIIOil)AL
IFAMII,IIES
M. I.(;I1.F,A Ihllt'fr~il~. Illm, fie llrlllvlb,s t' '~rh" #o~IM 2~1 ( fretful/ p!f,illt I/ I jl, Ilrudeldrd ,h* ilhm'pJl, ~ Jfi30 lJrl:xell~L Iil, IRftlll¢,
Ite'elved 2h FIdum!l) 11479 flllulle~.l~e t viral Im~ delhlvd i ~, ~, l I a m*t,r,fldal fmullv ,,r l!l,tphs rind ha< pmu d Ille e dq,.nu, i1{ t11111 ii1]nilltlfll btillilh!% called I,,, I!, i ~ ;Ira! ! , Ille! qL'l Ill lU'I}Ium~ [111 hinh/'ae III Itas qhl%{ll lhnl h,t eteiv ft. hllegeL il " 2, tllele If ft iiHllrlllllp; lmuilv ~f, fJ ,i Al~, I i Jl,I I¢ k lhe p0tl.u~e I,f thu ple%>lfl Illile ill hlllhl Zl ~el uf Illffltlllllffl frmfi,ie~ iiiillflltllrq~ rtll il,e M,, t, mid ith Illtilllh if ittlef ithtlufld*il lalfillle~
1, I~i~ltilitliilll
I;., ~, ~ti
tllllil:Oli~ ~; I~ it I~ilil lid, lit t,']l~ f~ if !~ Ft tlt!llt~ ~c,I ~ilid ,'b I~ it ~ilHl%:llull i,f "41h~lJh Ill I! h~ilt~d itltiTllfelld~lllf ~01tli ~t'fh th~ll fro, tl~l: It't ~if~ Uilftdlt~d z (tll tJt= Ji Ibt
l[ ,~Lf~
itlitl 7 ~: ~, Ihtsll i<~ #i
~lid, Ilfill I VUffll~
as.
A ~/lh~el ttf f7 llllf h~htitgillg In a ~. J~ ~1t~1 df~li~llfl#lll A s~ibs.vl ill i7 whicll i~ gll~lll~li0elll 1411~ ItlillJt11~II with !llis propl.~rty is called .~lig,ll. A rnillroJd M ctta ~lso he defilled ~s ii pair rE, 71 where ,<#' i~ a c(dleclicm Sllhae!s of E (cHImJ stiglllS) sttch lhat (a') [h'), It') are satisfied:
~i Ill
{a') ISl---1, w~,~; (h') If S iar, d S:~ are dPtinct rqerlbers c~f 7, then S i ¢ $2; (2') If S I and Sa are distinct mcmbe
1,14
M totea
~;, The ma~rotfl MtG, ~, b) I,el (; t~e a si;t~t~le graph, a be an inleger (t~ - I), b be an integer (()'~1)~2.), trod +~' be file edge sel of G, The function [ :,~l'd'i-~ N, defined hy
i'~ nen decreasing, stibmodaiar [3] and thus [9, Chap;er gl it is possible m obtain a matroid on Ihe edge set '~' of G, by taking us in:Icpendent sets the sets X ( X c ~) such that, v g c .xj, f ( Y I ~ [ Y l, This matroid is called M(G, a, h),
3° Charac~erizat~on of the ~ligms of M(G, a, b) I,et ~ be contained in ~, the edge set c,f G. Theorem 1. 9, i~ a stigm of M(G, a, b) if and only if (il a ]Us~:, Sb- b = ]9,1-1, and (ii) 9, is minimal non empty with property (i). It is enough to prove that, if 3° satisfies conditions (i) and OiL it is dependent (part I) :rod that, if 9, is a stigm, it satisfies condition (i) (part 2). Part I. Condition (1) implies ([(9,)<]9,1), which implies 9' is not independent in M(G, a, b I. Purl 2. Lcl ff be a stigm (if .a, is strictly contained in ,~, ~ is independent and f(.~)~> [.r~[; it is thus aecessary to have f(9,)<19,]), we have thus:
aI~UlS
b~[9,]-!,
(1)
:if [9,1 - l, as f(9,)= a [Us~:.~Sl-b>~O, (1) implies
and (i) is proved. If 19'[> 1, let 9,' be contained in 9,, with 19,'1= [9,[-1, ~ ' is independent, and thus f(9,') ~ 19,'i, and alsUpSl-b~>lg, '-1,
(2)
(I) and (2i imply
19ol =t~,.
I
U s =,~'>.i U ,1=t,>1:/'1~1,
tin I lhus (i) is proved,
4, ConnectMty M th:" stigma, a¢ M(G, 0, h) Theocem 2, bor every graph G, ]'or e~ery integer , (a-~ 1), for ,.cry nleger b (2a ~ b ~ (I), and for every sliqm 9Oof M(G, 0, b), tile partial graph r?, G ]orme'l
by the edges helanging to cp is connected, We shall prove this theorem by contradiction: let us suppose there art~ st (k ~ connected cempoeents in the partial graph generated by the edges hehmging the stigm .~. Let V, be the num~er of vertices al~d E, be the numher of edges the ith con ~ected component. A-, 9 o L, a stigm, every connected compon~!D[ independent in the matroid M ( O , a, b) and
aE-b~E,,
21 a; If is
i = 1 , 2 . . . . . k.
Thus
o
V;-kb~E
e,,
and
o i U S[-b~lgol+
5 M(G, a, b) and the mat~oid'd families of Simoes-Pereira and A~dreav If we fix a and b(a and b integer a ~ > l , ant; 2a~>b~>0), the s igms cf all the matroids M(G. a, b) are the members of a matroidal family ~ee note this matroidal family MF(a, b); of course, the elements of MF(a, b) or. characterized by the conditions (i) and (it) of Theorem I. In [5, 6, 7], Simoes-Pen.ira has proved the existence of four matroidal families called Fo, F2, F=, and F 1, the set of polygons, and in [1] Andreae has shown that. for every n integer, n ~>2, there is a
IOf~
M. Lorea
m a t r o i d a l f a m i l y M,, c o n t a i n i n g t h e c o m p l e t e g r a p h wi~dl n v e r t i c e s . It is n o t difficult to) s e e t h a t :
Fc~ = M 2 - M F ( I , 2); F I = M 3 - M F ( I , 1); F , = M F ( I , 0)
M,.-MJ'(½n,
(Bicircular graphs);
½r~+l),
n i n t e g e r />2,
n ~, :n;
and
~,1~- M F t ~ ( n
1), 1),
n i n t e g e r />3,
n odd.
Reterences [I} "1 /,ndlcae, M~llroidal families of finite connected nonhomeomorphic graphs exist, J. Graph "fhe,~ry 2 (197S~ 149-153 [2] C I-crg,: Graplv; and Hypergraphs (North Nollan]. Amsterdam, 1973) [~] M I.orga. t.a ,,tructure de terrains problemes d'aafeetetion. Rev, Frangaise Automal, Inf'~rmat Recherche Ophrati,mnelle, Sdrie R,O. 10 (1976) S9 6~ [~] M lorda Matroldes ,ur les ensembles d'ar&tes d'hyp,:rgrapheb, Cahiers Centre l~tudes Recherche Ophr. 20 (1978; 127 t3:, [ ~] J M S . Simoes-Pereira, Cn malroids on edge sets of glaphs with connected subg*aphs as circuits II, Discrete Math. 12 (1975) 55-78. [hi J M S Sim~ms-Pcruira. Subgraphs m, circuits and bases of matroias, Discrete Math. 12 (1975) 79-'~g. [ 7 ~M S. Simoe~ Pereira, A comment on matroidal families, Colloques Internationaux C N.R2k, No. 2(dl Probl~me,~ comhinatoires et tlahorie des graphe~,, Paris-Orsay 1976, i978, 385-387 {8] W [ulte Inlroduction to the Theory of Malroids, Modern Analylic and Computational Melhods in Scicn~ ,rod Mathematics 37 (Elsevier, Amsterdam, 197D, [9] D Welsh. Ma[rtlld Theory (Academic Press, b n d o n New York-San Frmlcisco, 1976).