Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases. A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$. The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ turns out to be the family of independent sets of a new matroid on $E$, called base-matroid and denoted by $M_B$. In this paper we prove that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti.  Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$

Maffioli, Francesco

Zagaglia Salvi, Norma

English

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 5, Number 1, Pages 1–6ISSN 1715-0868A CHARACTERIZATION OF THE BASE-MATROIDS OF AGRAPHIC MATROIDFRANCESCO MAFFIOLI AND NORMA ZAGAGLIA SALVIAbstract. Let M = (E,F) be a matroid on a set E, and B one ofits bases. A closed set θ ⊆ E is saturated with respect to B when|θ ∩B| = r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I ∩ θ| ≤ r(θ) for everyclosed saturated set θ turns out to be the family of independent setsof a new matroid on E, called base-matroid and denoted by MB . Inthis paper we prove that a graphic matroid M , isomorphic to a cyclematroid M(G), is isomorphic to MB , for every base B of M , if and onlyif M is direct sum of uniform graphic matroids or, in equivalent way, ifand only if G is disjoint union of cacti. Moreover we characterize simplebinary matroids M isomorphic to MB , with respect to an assigned baseB.1. IntroductionLet M = (E,F) be a matroid on a set E, having F as its family ofindependent sets. For notations and definitions we refer to [6].Let Ξ denote the set of all closed sets of M . ThenF = {S ⊆ E : |S ∩ θ| ≤ r(θ),∀θ ∈ Ξ}.A set θ ⊆ E is defined [3] saturated with respect to a base B of M if|θ ∩B| = r(θ).Thus any B-saturated closed set θ satisfies the relation cl(θ ∩ B) = θ; inother words, θ coincides with the closure of its intersection with B.If in addition θ belongs to Ξ, we have a saturated closed set. The set ofall the saturated closed sets of M , with respect to a base B, is denoted byΞB. A circuit is fundamental with respect to B when it is the fundamentalcircuit of an element i ∈ E \ B. Calling γ(i) the unique minimal subset ofReceived by the editors January 21, 2007, and in revised form September 29, 2008.2000 Mathematics Subject Classification. 05B35, 90C27.Key words and phrases. Base, uniform matroid, graphic matroid, binary matroid,cactus.Work partially supported by MIUR (Ministero dell’Istruzione, dell’Universita` e dellaRicerca).c©2010 University of Calgary12 FRANCESCO MAFFIOLI AND NORMA ZAGAGLIA SALVIB such that γ(i) ∪ i 6∈ F , then γ(i) ∪ i is a fundamental circuit. We use thenotationFB = {S ⊆ E : |S ∩ θ| ≤ r(θ), ∀θ ∈ ΞB}andMB = (E,FB).In [3] it is proved that M = (E,FB) is a matroid, and in particu-lar a transversal matroid. An application of these matroids, named base-matroids, is in the field of inverse combinatorial optimization problems;indeed many different inverse problems have been addressed in the recentliterature [1, 3, 5].Recall that a matroid M on a ground set E, whose family of independentsets is F , is direct sum of the matroids M1,M2, . . . ,Ms on disjoint setsE1, E2, . . . , Es respectively, when E1, E2, . . . , Es is a partition of E andF = {I1 ∪ · · · ∪ Is : Ii ∈ F(Mi), 1 ≤ i ≤ s},where F(Mi) is the family of independent sets of Mi.A simple matroid M is binary if the symmetric difference of any twodifferent circuits is a union of disjoint circuits. Clearly graphic matroids areexamples of binary matroids.The main aim of this paper is determining a characterization of a graphicmatroid M which is isomorphic to MB (M ' MB), where B is any base ofM . Indeed, it is proved that a matroid M , isomorphic to a cycle matroidM(G), is isomorphic to MB for every base B of M if and only if G isdisjoint union of cacti or, in equivalent way, if and only if M is direct sum ofuniform graphic matroids. Finally we characterize a simple binary matroidM isomorphic to MB, with respect to an assigned base B.2. Independent circuitsLet F and FB denote the collections of independent sets of M and MBrespectively. It is easy to see thatF ⊆ FB,and the inclusion is proper when a dependent set of M turns out to beindependent in MB; in this case M is not isomorphic to MB. In other wordsthe above relation implies that M 'MB if and only ifF = FB.Lemma 2.1. Let M be a matroid and B one of its bases. Then M 'MB ifand only if every circuit of M is also circuit of MB.Proof. If every circuit of M is also circuit of MB, then it follows that everydependent set of M is dependent also in MB. Then F = FB and conse-quently M 'MB.Conversely, if M ' MB, from the condition F ⊆ FB it follows F = FB.Then it is not possible that there exists a dependent subset of M whichturns out to be independent in MB. A CHARACTERIZATION OF THE BASE-MATROIDS OF A GRAPHIC MATROID 3We first consider the case of a circuit of M , dependent in MB.Proposition 2.2. Assume that a circuit C of M satisfies the inequality|C ∩ θ| > r(θ) for a suitable closed set θ of M saturated with respect to abase B. Then θ = cl(C).Proof. There are two cases to consider depending on the condition that Cis not contained or contained in θ.If C is not contained in θ, then C ∩ θ is a proper subset of C; thenit is independent in M and consequently independent also in MB. Thus|C ∩ θ| ≤ r(θ), a contradiction.In the second case, we have |C ∩ θ| = |C|; then r(C) ≤ r(θ). As r(C) =|C|−1, we obtain the following double inequality |C|−1 ≤ r(θ) < |C|. Thenr(θ) = |C| − 1 and therefore θ = cl(C). Definition 2.3. A circuit C of M is said to be independent with respect toB, or B-independent, if|cl(C) ∩B| < |C| − 1.Moreover C is dependent with respect to B, or B-dependent, if it is notindependent with respect to B; that is,|cl(C) ∩B| = |C| − 1.Thus cl(C) is saturated with respect to B.Notice that if a circuit C is B-dependent, then C /∈ FB. In other wordsC is dependent in MB; in particular it is a circuit of MB. On the contrary,if C is B-independent, then C is independent in MB and consequently M isnot isomorphic to MB.Recall ([2]) that a circuit C of a matroid M has a chord e if there are twocircuits C1 and C2 such that C1 ∩ C2 = {e} and C = C1 4 C2. In this casewe say that C is the sum of C1 and C2 and also that C ∪ {e} is split intoC1 and C2.When a chord belongs to a base B, we say that it is a B-chord.Lemma 2.4. A circuit of M , fundamental with respect to B, is B-dependentand does not contain B-chords.Proof. Let C be a circuit ofM fundamental with respect toB. If |C| = m+1,then |C ∩ B| = m and C is B-dependent. If C contains a B-chord e, thencl(C) containsm+1 elements which belong to B. This implies the impossiblerelation r(cl(C)) = m+ 1. Proposition 2.5. Let M be a uniform matroid of rank n. Then for everybase B of M it is M 'MB.Proof. Let C be a circuit of M , that is a (n+ 1)-subset of E(M). It followsthat |C ∩ E| > r(E), so that C is dependent also in MB. It is in partic-ular a circuit because every proper subset of C is independent in M andconsequently in MB. The result follows from Lemma 2.1. 4 FRANCESCO MAFFIOLI AND NORMA ZAGAGLIA SALVI3. Graphic matroidsIn this section we consider the problem of characterizing graphic matroidsM isomorphic to MB for every base B of M . Let G = (V,E) be a graphwithout loops and parallel edges, having V and E as the sets of vertices andedges respectively.Recall that two cycles of a graph are said intersecting when the intersec-tion of their edge sets is not empty.Lemma 3.1. A cycle matroid M(G), having rank n, is uniform if and onlyif G is either an n-tree or an (n+ 1)-cycle.Proof. Let us assume that M is uniform. If m is the number of edges ofG, then either m = n or m > n. In the first case M(G) does not containdependent sets; then G does not contain cycles and G is an n-tree. If m > n,the condition that M is uniform implies that every (n + 1)-subset forms aminimal dependent set, that is a (n+1)-cycle of G. Let C be a (n+1)-cycleand e = (u, v) a possible edge of E \ C. Then u and v can not belong toC because otherwise we obtain a chord of C and then a cycle having lengthlesser than n+1. Thus at least one of the vertices u and v does not belong toC; this implies that there exists a spanning tree having cardinality greaterthan n, a contradiction.Conversely, if G is either an n-tree or an (n + 1)-cycle, then in both thecases M(G) has rank n. In the first case it is a free matroid, while in thesecond case it is the uniform matroid Un,n+1. Lemma 3.2. Let G be a graph having two intersecting cycles; then G con-tains two cycles C and H such that C 4 H is one cycle and C ∩ H is apath.Proof. Let C and Q two intersecting cycles; C4Q is a set of disjoint cycles.Let D one of these cycles, where D = C ′ ∪ H ′, C ′ ⊆ C and H ′ ⊆ Q arepaths and H ′ is vertex disjoint from C ′, but on the end vertices.The subgraph C 4D coincides with (C \ C ′) ∪H ′. In other words, it isobtained from C by replacing the path C ′ by the path H ′. Then C 4D isone cycle and C ∩D = C ′. Proposition 3.3. Let G be a graph having two intersecting cycles. ThenM(G) contains a base B in relation to which MB is not isomorphic to M .Proof. Let G be a graph having two intersecting cycles, say C and H. ByLemma 3.2 we may assume that C4H is one cycle, say D, and C ∩H = Pis a path of length ≥ 1. Assume that D = C ′ ∪H ′ where C ′ , H ′ are pathscontained in C and H, respectively.Let B a spanning tree of C ∪ H obtained by taking all the edges of Cbut an edge e of P and all the edges of H but e and another edge, say f , ofH \P . We may extend B to a spanning tree of G , which we still denote B.Then we may see that H is not B-fundamental because contains two edgesA CHARACTERIZATION OF THE BASE-MATROIDS OF A GRAPHIC MATROID 5which do not belong to B. Then|cl(H) ∩B| = |H| − 2and H is B-independent. This implies that M(G) is not isomorphic to MB,with respect to the base B. Recall that a connected graph G is called a cactus when any edge belongsto at most one cycle. In other words G is a cactus if and only if it isconnected and its possible cycles are edge-disjoint.Corollary 3.4. If the cycle matroid M(G) is isomorphic to MB for everyspanning tree B of G, then G is a graph whose components are cacti.Proof. From Proposition 3.3 it follows that G has not intersecting cycles; inother words the components of G are cacti. Theorem 3.5. A cycle matroid M(G) is isomorphic to the base-matroidMB, for every base B of M , if and only if G is a disjoint union of cacti.Proof. If a cycle matroid M(G) is isomorphic to the base-matroid MB, forevery base B, then, by Proposition 3.3, G does not contain intersectingcycles and by Corollary 3.4 the components of G are cacti.Conversely, if the components of G are cacti, then G has not intersectingcycles. If there exists a base B in relation to which MB is not isomorphicto M , then, by Lemma 2.1, there exists a cycle Q of G, which turns out tobe independent in MB. Clearly by Lemma 2.4 Q is not fundamental withrespect to B. Denote by f an element of Q \B; then the fundamental cycleF (f), obtained by adding f to B, and Q are distinct and intersecting, acontradiction. Theorem 3.6. For every base B of a graphic matroid M , M 'MB if andonly if M is direct sum of uniform graphic-matroids.Proof. Let M = ⊕Mi be direct sum of uniform graphic matroids and B abase of M . The B = ⊕Bi, where Bi is a base of Mi. By Proposition 2.5Mi 'MiBi and therefore M 'MB.Now assume that M is isomorphic to the cycle-matroid M(G) and more-over that M 'MB in relation to a base B of M , that is a spanning tree ofG. Then by Theorem 1 G is union of disjoint cacti and therefore does notcontain intersecting cycles. This implies that E(G) can be partitioned intoedge-disjoint cycles, say C1, C2, . . . , Cr, r ≥ 0, and edge-disjoint trees, sayT1, T2, . . . , Ts, s ≥ 0. Then M is direct sum of the matroids on C1, C2, . . . , Crand T1, T2, . . . , Ts, which turn out to be all uniform.Thus M(G) is direct sum of uniform graphic matroids. Now we generalize the result of the previous theorem to the case of asimple binary matroid.Theorem 3.7. Let M be a simple binary matroid on E and B a base of M .Then M 'MB if and only if either all the circuits of M are fundamental or6 FRANCESCO MAFFIOLI AND NORMA ZAGAGLIA SALVIevery circuit not fundamental with respect to B contains at least one chordwhich belongs to B.Proof. If M ∼= MB, then by Lemma 2.1 every circuit of M is also a circuitof MB; in other words every circuit of M has to be B-dependent. Let C bea n-circuit, not fundamental with respect to B. Because it is B-dependent,then |cl(C) ∩B| = n− 1.From the condition that C is not fundamental it follows there exists atleast an element, say a, which belongs to (cl(C) \ C) ∩ B. Because M isbinary, from the proof of Lemma 2.1 of [2], it follows that every element ofcl(C) \ C is a chord; then the element a is a B-chord.Conversely, assume that every possible circuit, not fundamental with re-spect to B, contains at least one B-chord. Our aim is to prove that it isB-dependent; by Lemma 2.1 this implies that M ∼= MB. Let C be an n-circuit, not B-fundamental, having a B-chord, say c1. Let H1, H2 be twocircuits in which C ∪ c1 is splitted. If H1 and H2 are both B-fundamental,then (C ∩ B) ∪ c1 is an independent set of cardinality n − 1 whose closurecoincides with cl(C). Then C is B-dependent.Now, assume that at least one of the above circuits, say H2, is not B-fundamental. Then it contains at least one chord c2 which belongs to B,such that H2 ∪ c2 can be decomposed into two distinct circuits intersectingin c2. By repeating the above procedure, we arrive to obtain that C can bedecomposed into a number, say s, of fundamental circuits. Thus C contains selements which do not belong to B and s−1 chords which belong to B. If T isthe set of similar chords, then |cl(C)∩B| = |(C∩B)∪T | = n−s+s−1 = n−1and C is still B-dependent. References1. M. Cai, Inverse problems of matroid intersection, J. Comb. Optim. 3 (1999), no. 4,465–474.2. R. Cordovil, D. Forge, and S. Klein, How is a chordal graph like a supersolvable binarymatroid?, Discrete Mathematics 288 (2004), 167–172.3. M. Dell’Amico, F. Maffioli, and F. Malucelli, The base-matroid and inverse combina-torial optimization problems, Discrete Appl. Math. 128 (2003), 337–353.4. J. Edmonds and D. R. Fulkerson, Transversals and matroid partition, J. Res. Nat. Bur.Standards Sect. B 69B (1965), 147–153.5. C. Heuberger, Inverse combinatorial optimization: A survey on problems, methods andresults, J. Combin. Optim. 8 (2004), no. 3, 329–361.6. J. G. Oxley, Matroid theory, Oxford University Press, New York, 1992.Dip. di Elettronica e Informazione, Politecnico di MilanoP.za L. da Vinci 32, 20133 Milano, ItalyE-mail address: maffioli@elet.polimi.itDip. di Matematica, Politecnico di MilanoP.za L. da Vinci 32, 20133 Milano, ItalyE-mail address: norma.zagaglia@polimi.it

A characterization of the base-matroids of a graphic matroid

https://cdm.ucalgary.ca/article/download/61894/46637

A characterization of the base-matroids of a graphic matroid

Abstract

Similar works

Full text

Available Versions

Contributions to Discrete Mathematics (E-Journal, University of Calgary)