- Email: [email protected]
Independence systems with continuous cardinality of bases
Discrete Mathematics 31 (1980) 107-109 @ North-Holland Publishing Company
COMMUNICATION
INDEPENDENCE SYSTEMS WITH CONTINUOUS CARDINALITY OF BASES D. NADDEF,
N. SBIHI
and M. TCHUENTE
Laboratoire I.M.A.G., BP. 53X, 38041 Grenoble, France Communicated by C. Benzaken Received 11 February 1980
1. Intrsduction The maximum cardinality basis problem in a genoid (i.e. an independence system), although NP-complete in general, is solvable in a polynomial time in the following cases: matchings of a graph [l], intersection of two matroi’ds [2,4] and independent set of vertices in claw-free graphs [S, 71. These three genoi’ds satisfy the continuous cardinal&y of bases for all their minors. The aim of this paper is to characterize those genoids having this property and to answer thereby a question posed by Sakarovitch [6].
2. A few definitions Definition 1.Let E be a finite non empty set and 9 a family of subsets of E such that
The pair % = (E, 9) is called a genoi’d on E. We call independent sets of % the elements of 9, buses of % the maximal (by inclusion) elements of 9 and circuits of Y# the minimal elements fo P(E) - S. A genoid can be defined by the set of its bases or as well by the set of its circuits. Definition 2. Let 3 be a gendid on E defined by the set % of its circuits and S a subset of E. We define the contraction of 9 over s’ (respectively the reduction of % 107
D. Naddef, N. Sbihi, M. Tchuente
108
:o S)
by 9 x S (resp. ($9 S), with circuit family
zznoi’d, denoted
to be the
l
u(sexs)={cE%~C~S} (req. %(% . S) = min(F# 8, F = C n S and CE %I). A minor of $ is any gendid obtained from 29 by a finite nur&er of contractions and reductions. It can be shown using the methods of Tattc [S] that any minor of % is of the form: (3~ S). T where Tc S c E. DeisnitioIB 3. A geno’id defined by the family 92 of its bases satisfies the pcontinuous cardinal& of hoses property (p E N”) if: WB E 92 (B not of maximum cardinal&y), 3 B’E 9 such that 0 < IS’\ - [BIG p. 3. The 4hewe.m
Theorem. All the minors of a genoi’d98satisfy the p-continuous cardinality of bases propetiy if and only if % has no minor whose circuit family is isomorphic to the edge set of &.tp+2p0 Note, This theorem has been proved by F. Jaeger [3] in the case of the genoi’d of
independent
sets of vertices for p = 1.
Prod of the theorem. (;,J Necessary condition; trivial, (b) sufficient conditioll: Let 93 be a gendid satisfying the condition
of the theorem and B a basis OC% not of maximum cardinality. Consider a basis B’ of % *such that 1B’l>IBf and IS n B’( is maximum. Let a E B -- B’, %’= % x (8’ U {a)) and %’ the circuit family of %‘. Any element C of %?’cannot be contained in B, so C n (B’- B) # 8. Therefore let 0 be a minimal transversal of %’ contained in B’- B. By minimality of In, we have: WoEl2,
ICE%’
such that
Since every circuit of %’ contains
W%’
l
nlqJ+
a, it follows that
= {{a, W}/OE a).
(QU(a)))
Thus %([%X(B’U(a})] hyrathesis, we have
Cn0={u}.
l
(i2 W(a)))
is isomorphic
to the edge set of Kl,lnl. By
1.
B’U{a}-0
is an independent set of 99 and since \777BI>IB’nBI, then by the maximality of !B’nBl we Ina~ ITI~IB~ so
Since T=
B’[+ l-I.ni6IBl hence
Independence systems
109
[l] J. Edmonds, Paths, trees and flowers, Canad. J. Math. 17 (1965) 449-67. [2] J. Edmonds, Matro’id intersection (Conference on Discrete Optimization, Vancouver, August 1977), Annals Discrete Math., Vol. 4 (North-Holland, Amsterdam, 1979) 39-49. [3] F. Jaeger, Etude de quelques invariants et problemes d’existence en theorie des graphes, These d’Etat, Universite Scientifique et Medicale de Grenoble (1976). [4] E. Lawler, MatroId intersection algorithms, Math. Programming 9(l) (1975) 31-56. [S] G.J. Minty, On maximal independent sets of vertices in claw-free graphs, Department of Mathematics, Indiana University (1977). [6] M. Sakarovitch, Deux ou trois chases que je sais des ‘bitrdides, Rapport de recherche, I.M.A.G. (1976). [7] N. Sbihi, Algorithme de recherche d’un stable, de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980) 53-76. [S] W.T. Tutte, Lectures on matrdids, J. Research NBS 69B (1) (1965).
COMMUNICATION
INDEPENDENCE SYSTEMS WITH CONTINUOUS CARDINALITY OF BASES D. NADDEF,
N. SBIHI
and M. TCHUENTE
Laboratoire I.M.A.G., BP. 53X, 38041 Grenoble, France Communicated by C. Benzaken Received 11 February 1980
1. Intrsduction The maximum cardinality basis problem in a genoid (i.e. an independence system), although NP-complete in general, is solvable in a polynomial time in the following cases: matchings of a graph [l], intersection of two matroi’ds [2,4] and independent set of vertices in claw-free graphs [S, 71. These three genoi’ds satisfy the continuous cardinal&y of bases for all their minors. The aim of this paper is to characterize those genoids having this property and to answer thereby a question posed by Sakarovitch [6].
2. A few definitions Definition 1.Let E be a finite non empty set and 9 a family of subsets of E such that
The pair % = (E, 9) is called a genoi’d on E. We call independent sets of % the elements of 9, buses of % the maximal (by inclusion) elements of 9 and circuits of Y# the minimal elements fo P(E) - S. A genoid can be defined by the set of its bases or as well by the set of its circuits. Definition 2. Let 3 be a gendid on E defined by the set % of its circuits and S a subset of E. We define the contraction of 9 over s’ (respectively the reduction of % 107
D. Naddef, N. Sbihi, M. Tchuente
108
:o S)
by 9 x S (resp. ($9 S), with circuit family
zznoi’d, denoted
to be the
l
u(sexs)={cE%~C~S} (req. %(% . S) = min(F# 8, F = C n S and CE %I). A minor of $ is any gendid obtained from 29 by a finite nur&er of contractions and reductions. It can be shown using the methods of Tattc [S] that any minor of % is of the form: (3~ S). T where Tc S c E. DeisnitioIB 3. A geno’id defined by the family 92 of its bases satisfies the pcontinuous cardinal& of hoses property (p E N”) if: WB E 92 (B not of maximum cardinal&y), 3 B’E 9 such that 0 < IS’\ - [BIG p. 3. The 4hewe.m
Theorem. All the minors of a genoi’d98satisfy the p-continuous cardinality of bases propetiy if and only if % has no minor whose circuit family is isomorphic to the edge set of &.tp+2p0 Note, This theorem has been proved by F. Jaeger [3] in the case of the genoi’d of
independent
sets of vertices for p = 1.
Prod of the theorem. (;,J Necessary condition; trivial, (b) sufficient conditioll: Let 93 be a gendid satisfying the condition
of the theorem and B a basis OC% not of maximum cardinality. Consider a basis B’ of % *such that 1B’l>IBf and IS n B’( is maximum. Let a E B -- B’, %’= % x (8’ U {a)) and %’ the circuit family of %‘. Any element C of %?’cannot be contained in B, so C n (B’- B) # 8. Therefore let 0 be a minimal transversal of %’ contained in B’- B. By minimality of In, we have: WoEl2,
ICE%’
such that
Since every circuit of %’ contains
W%’
l
nlqJ+
a, it follows that
= {{a, W}/OE a).
(QU(a)))
Thus %([%X(B’U(a})] hyrathesis, we have
Cn0={u}.
l
(i2 W(a)))
is isomorphic
to the edge set of Kl,lnl. By
1.
B’U{a}-0
is an independent set of 99 and since \777BI>IB’nBI, then by the maximality of !B’nBl we Ina~ ITI~IB~ so
Since T=
B’[+ l-I.ni6IBl hence
Independence systems
109
[l] J. Edmonds, Paths, trees and flowers, Canad. J. Math. 17 (1965) 449-67. [2] J. Edmonds, Matro’id intersection (Conference on Discrete Optimization, Vancouver, August 1977), Annals Discrete Math., Vol. 4 (North-Holland, Amsterdam, 1979) 39-49. [3] F. Jaeger, Etude de quelques invariants et problemes d’existence en theorie des graphes, These d’Etat, Universite Scientifique et Medicale de Grenoble (1976). [4] E. Lawler, MatroId intersection algorithms, Math. Programming 9(l) (1975) 31-56. [S] G.J. Minty, On maximal independent sets of vertices in claw-free graphs, Department of Mathematics, Indiana University (1977). [6] M. Sakarovitch, Deux ou trois chases que je sais des ‘bitrdides, Rapport de recherche, I.M.A.G. (1976). [7] N. Sbihi, Algorithme de recherche d’un stable, de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980) 53-76. [S] W.T. Tutte, Lectures on matrdids, J. Research NBS 69B (1) (1965).