MATROID COMPLEXES–GEOMETRICAL REPRESENTATIONS OF MATROIDS

5 (1985), 1, 85-42

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MATROID COMPLEXES-GEOMETRICAL REPRESENTATIONS OF MATROIDS* Liu Guizhetn (j"lJ ;ff • ) Department of Mathematics, Shandcmg University, Jinan, China

Abstract In this paper we introduce a new method to represent matroids by complexes. Such a complex is called a matroid complex. Necessary and sufficient conditions for a complex to be a matroid complex are given and certain concepts in matroid theory are extended to complexes, so that one may solve certain problems concerning matroids by converting them into problems concerning complexes. At the end of this paper some interesting open problems are proposed.

§ 1. Introduction Whitney[lJ firstly put forth the theory of matroids in 1935 and discussed the relations between matroids and m'atrices. Thence many people studied the methods of representing matroids. Ingleton[2J discussed the vectorial representation of matroids. Holzm(JfJ1ffb[3J, Donald and others introduced the concept of the base graphs of matroids. Mason[4J discussed the geometrical representation of matroids. In this paper matroids are represented by geometric complexes in Euclidean space. This method establishes connections between complexes and matroids so that the study of certain properties of matroids may be converted into the study of the geometrical properties of complexes. Some concepts of matroids are extended to n-complexes. The homology theory of complexes can be introduced into the theory of matroids.

§ 2. Geometrical Representation of Matroids In this paper by complexes we shall mean the geometric complexes ~

• Received 11 Sept., 1982.

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unless state otherwise. We make the convention that the empty set 0 is considered as a simplex of every complex. We use the same notation to denote a simplex and the set of its vertices. If a is a vertex of a simplex A, then we write a E A. We denote by IA I the number of vertixes of the simplex A and by Er(k) (O~'1'~n) the collection of r-etmplexes of the fl,-eomplex k. A matroid M is a finite set E and a collection I of subsets of E such that 1° and 2° are satisfied. 1° 0EI and if XEI, YcX, then YEI. 2° If X, YEland [Y!>IXI then:3eEY\XsuchthatXU{e}EI. Write M = (E, I). Each menber of I is called an independent set of the matroid M = (E, I). A subset of .E not belonging to I is called a dependent set. A circuit of M is a minimal dependent set. The rank function of a matroid is a function p: 2E~Z defined by p(A)=max{IXIIXcA, XEI}(ACE) where 2E denotes the collection of all subsets of E and Z denotes the set of all non-negative integers. The rank of the matroid M is defined to be the rank of the set E. A loop of M is an element e of E such that {e} E1. A matroid without loops is called a normal matroid. Definition 2.1. Let M = (E, I) be a matroid and K be a geometric complex. If there exists a bijection f: E~Eo(K) mapping E onto Eo(K), such that for any AcE, AEI if and only if f(A) is the set of vertices of a simplex of the complex K, then we say that K is a representive complex or a geometrical representation of the matroid M. Proposition 2.1. There exists a representive complex of a matroid M = (E, I) if and only if M is a normal matroid. Proof. Let M = (E, I) has a representive complex K. By Definition 2.1 there exists a bijection f: E~Eo(K) such that for any eEE, fee) EEo(K) Le., {fee)} is the set of vertices of the simplex of K. So {e} EI, it follows that M is a normal matroid. Now, suppose that M = (E, I) is a normal matroid and E={el, e2, ''', e,,}. Thus for VeEE, {e}EI. We construct an abstract complex I){ such that E is the set of its vertices and VX EI is the set of vertices of an abstract simplex of I){. Let M have rank n+1, then I){ is an abstract n-eomplex. We have known that an abstract n-complex I){ has a geometrical realization K in Euclidean (2n+1)-space. Let Eo(K) = {al' 0,2, '::» a,,} be the set of vertices of K, where each 0,1 corresponds to el(i=1,2, "', k). It is clear that the bijection f: el~al (i=1, 2, "', k) satisfies the conditions of Definition 2.1, i.e., K is the representive complex of M. I

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It is easily Seen that a normal matroid has a rank <2 if and only if its representive complex is a 1-complex, I.e., a graph. Example. We have known that the smallest non-graphic matroid is the uniform matroid U2,4= (E, 1) where E={1, 2, 3, 4} and 1={XIXcE,IXI<2}[5J. Its representive complex is shown in Fig. 1. Its base graph is shown in Fig. 2. {1,2}

1

2 .a::---J---~-~

{l, 3}

{2, 4} K--~-----t'--------P {2, 3}

3

4

Fig. 1

{3,4}

Fig. 2

For an abnormal matroid it has no representive complex. But it may be represented as follows: . Definition 2.2. Let M = (E, 1) be a given matroid and E= {eo, e1, "', 6],J. Let El<= (co, C1, "', Cl<) be a k-simplex and K be a suboomplex of El<. If there exists a bijection f: ei~cl<,(i=1, 2, "', k) mapping E onto the set of vertices of El< such that for any XcE, XE1, if and only if f(X) is the set of vertices of the simplex of K, then the pair (El<, K) is called a geometrical representation of M. I Obviously,according to this definition any matroid has a geometrical representation in Euolideoo space. If M is not a normal matroid, then the vertex 0, corresponding to a loop of M is not in K, and conversely.

§ 3. Matroid Complexes We recall that if K is a n---complex and each of its simplexes is a face of a n-simplex of K, then K is called a pure n-complex[6J. We also call it pure. At first let us extent some concepts in graph theory to complexes as follows: Let K be a n-complex and A, BE E tl (K). If there exists a finite number of simplexes Al,=A, Ai" "', A'j=B such that any two adjacent ones of them have a common (n-1)-dimensional face, then we say that

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there exists the n-dimensional path from A to B. If, for any A and BEEn(K), there exists the n-dimensional path from A to B, then K is said to be n-dimensionally eormeoted'P, Definition 3.1. A n-complex K is said to be strongly ndimensionally connected, if K is pure and n-dimensionally connected. Definition 3.2. Let K be a n-complex and ScEo(K). We define the induced subcomplex K [SJ of S in K to be the subeomplex of K which has as its simplexes just those simplexes of K each of which has vertices all belonging to S. Definition 3.3. Let K be a n-complex. If for any two simplexes A, B of K with dim A
Theorem 3.2.

I

A n-complex K is a representive complex of a matroid if and only if for any SCEo(K) and S =I=(/J, K [SJ is a pure complex. Proof. Let K be a representive complex of a matroid M = (E, I). We are to prove that for any S~Eo(K) and S=I=(/J, K[SJ is a pure complex. Assume the contrary, suppose that there is a set So=l=(/J and So~Eo(K) such that K [SoJ is not pure, and K [SoJ is a r-complex, then 1'>0. Let D = {A [A is a simplex of K [SoJ and is not a face of any r-simplex of K[So]}, then D=I=(/J. There is a set A o ED such that lAo 1 = max 1A I· By the definition of D it is clear that there is AED

BE Er(K [SJ) with dim Ao
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K is a matroid complex. Sinoe Ao and B are simplexes of K [SoJ, thev are also simplexes of K. From Definition 3.3 it is obvious that there is bE B\A such that A o U {b} is the set of vertioes of a simplex of K. Because A o and {b} are simplexes of K[SoJ and A o is not a face of a r-simplex of K [SoJ, Ao U {b} is not a face of a r-simplex of K [SoJ and A o U {b} is also a simplex of K [SoJ. Hence A o U {b} ED. But lAo U {b} I> lAo I· This oontradiots the choice of A o• Conversely, let A and B be any two simplexes of K such that IA I< IB I. Let S = A U B, obviously S~Eo(K), and S+f/J. By the hypothesis K[SJ is pure. Suppose that K [SJ is a r-oomplex. Thereby any simplex of K [SJ is a face of a r-simplexof K[SJ. Sinoe IAIA. Thus 3aoEBnO\A such that AU {ao} is a simplex of K [SJ. Certainly it is also a simplex of K and aoEB\A Le., K is a matroid complex, According to Lemma 3.1 it follows that K is the representive complex of a matroid. I Lemma 3.3. A n-oomplex K is a representive complex of a matroid if and only if K is a pure n-oomplex and for VA", BnEE,,(K) and A"+ B" if A,,-l is any (n-l)-dimensional face of A" then 3ajEBn such that A,,-l U {a j } constitutes a n-simplex of K. It can be proved by Definition 3.1 and Lemma 3.1. i Oorollary. If M is a normal matroid with rank n+l and K is a representive complex of M, then K is strongly 'llr-dimensionally connected. Proof. By Theorem 3.3 K is pure. Let A", BnEE,,(K) and A"+Bn we are to prove that there exists a n-dimensional path from A" to B" by induction on IA"\Bn I. The result is obviously true when IA"\Bn I = 1. So we Suppose that the corollary holds when IA"\BnI =m-l (l~m-l
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known from Theorem 3.2 that for VSCEo(K) and S +0, K [S] is pure. By K [S] and tr instead of K and n repeating the proof of corollary of Lemma 3.3. We shall have that K[S] is strongly n-dimensionally connected. Conversely, for VSCEo(K) and S+0 if K[S] is strongly r-dfmenslonafly connected, then it is obvious from Definition 3.1 that K [S] is pure. By Theorem 3.2 it follows that K is a representive complex of a matroid. I Let B be a simplex of the complex K. B is called a maximal simplex of K if it is not a proper face of any simplexes of K. Let E(K) be the collection of all maximal simplexes of K. For any SCEo(K), we denote the dimension of K [S] by R(S), where R(S)={max{!XI-l jXCS, XEE(K)}, -1,

S+0, S=0,

R(S) is also called the upper dimension of K [S] . Let min { IX I - l IX C S , XEE(K)}, S+0, ".(S) = { -1, S=0, "'(8) is called the lower dimension of K [S]. By Theorem 3.2 it is easily seen that the following theorem is true. Theorem 3.5. An-complex K is a representive complex of a matroid if and only if for any SCEo(K), R(S) =".(S). I

§ 4. Generalization of Some Basic Concepts of Matroids to Complexes At first we extend Some concepts of graphs to complexes. Let K be a n-complex and SCEo(K). If K(S) is a '1"-complex (-l~".~n), but for VaEEo(K)\S, K[SU{a}] is a (".+l)-complex, then K [S] is called a maximal ".-complex of K. Definition 4.2. Let K be a n-complex and SCEo(K), S is called a n-covering of K if every n-simplex of K has at least one vertex belonging to S. If S is a n-covering of K but for VS'eS, S' is not a n-covering of K, then S is called a minimal n-covering. Definition 4.3. NcE,,(K) is called a n-matching of K, if for VA, BEN, AnB=0· Let K be a matroid n-complex. It is known from the proof of Lemma 3.1 that there is a matroid M(K) =M(Eo(K), I) where 1= {A IA is the set of vertices of a simplex of K}. It is not difficult to prove the following propositions.

Definition 4.1.

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1. SCEo(K) is an independent set of M (K) if and only if S is the set of vertices of a simplex of K. Especially SCEo(K) is a base of M(K) iff S is the set of vertices of a n-simplex of K. 2. For VSCEo(K). Let peS) be the rank: function of M(K), then peS) = the dimension of K [S] + 1 = R(S) + 1. 3. SCEo(K)iS a flat l5J of M (K) if and only if K [S] is a maximal subcomplex of K. 4. OCEo(K) is a circuit of the matroid M(K) if and only if K[S] is a boundary su boomplex'f . 5. HCEo(K) is a hyperplane''" of M(K) if and only if K[H] is a maximal (n-l)-subcomplex of K. 6. O*cEo(K) is a eocircuit of M(K) if and only if 0* is a minimal n-oovering of K. We can deduce theorems about relations between the matchings and the coverings of a complex. But we shall not go into details of them. The concept of dual matroids may be extented to complexes Definition 4.4. Let V be the set of vertices of a k-simplex E"k = (ao, ai, "', a"k) and K be a suboomplex of E", A suboomplex K* of E"k is called the dual complex of K in E"k, if the collection of its maximal simplexes is E(K*) = {A*! (V\A*) EE(K)}. It is easily seen that (K*)* =K in E"k and if K is a matroid complex, then K* is also a matroid complex. Thus if (EIe, K) is a geomatrical representation of a matroid M in Euclidean space, then (EIe, K*) is that of the dual matroid M* of M. Relations between the suboomplexes of complexes and the minors of matroids can also be established. But we shall not explain in detail.

§ 5. Further Problems By the use of the concept of matroid complexes algorithms in matroid theory may be used to solve some combinatorial problems of complexes. For example, Edmoru1s' algorithm can be used to find the matching of a complex'P, On the other hand, the homology theory of complexes can be introduced into the theory of matroids. Let M be a normal matroid with rank n+l and K be a representive complex of M. The If-chain group, cyclic group, boundary group and homology group of K can be thought as that of M, respectively (O~If~n). Betti number and EuZelf number of M can be defined by the previous method. Especially, EuZ61f number n+i

X( M) of M is equal to ~ ( -l),-li" where i, is the number of independent r=l

sets whioh oontains If elements. Thus some extremal problems of matroids

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can be converted into those of complexes. For instance, Welsh's oonjeeture-" i7<;;;;;'min{i7<;-l, i7<;+1}(2
References [1] [2] [3] [ 4] [5] [6] [7] [8]

Whitney, H., Amer. J. Math. 57 (1935), 509-533. Ingleton, A. M., Combinatorial Mathematics and its Applications, Academic Press (1971), 149-169. Holzmann, C. A., Norton, P. G., and Tobey, M. D., SIAM J. Appt. Math. 25(1973), 618-627. Mason, J. H., Froc. Amer. Math. Soc. 30:1 (1971), 15-21. Welsh, D. J. A., Matroid Theory, Academic Press, London, New York (1976),7-9. Jiang, Z. H., Introduction to Topology, Shanghai Publishing House of Science and Technology(1978) (in Chinese). Xie, L. T., Nath», Sci. J. Shandong Univ. 1 (1980), 1-11 (in Chinese). Welsh, D. J. A., Combinatorial Mathematics and its Applications, Academic Press (1971), 291-307.