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Note on binary simplicial matroids
Discrete Mathematics 49 (1984) 105-107 North-Holland
105
NOTE
NOTE ON BINARY SIMPLICIAL MATROIDS
Mathematical Institute, Czechoslovak Academy of Sciences, ~itnd 25, 115 67 Prague 1, Czechoslovakia Received 13 January 1983 Using an earlier characterization of simplicial hypergraphs we obtain a characterization of binary simplicial matroids in terms of the existence of a special base.
1. Introduction W e shall assume basic knowledge of matroid theory. For undefined notions, see [5]. Let B be a base of a matroid M =" (E, qg), where ~ denotes the set of circuits of M. For any a ~ E \ B there is a unique circuit Ca - B tA{a}. T h e circuit Ca is called the fundamental circuit associated with B and a. A matroid M is binary iff, for any base B of M, every circuit of M is a symmetric difference of fundamental circuits associated with B. H e n c e any binary matroid is uniquely determined by the set of fundamental circuits associated with any base. For a binary matroid M = (E, q~) and a base B of M we define the fundamental hypergraph associated with B to be the hypergraph (B,{Caf'IB: a c E \ B } ) . T h e fundamental hypergraph can have e m p t y edges (corresponding to loops of M), o n e - e l e m e n t and multiple edges (both corresponding to parallel elements of M). Since any matrix o v e r GF(2) is the incidence matrix of a hypergraph, we have the following
Lemmua 1. A n y hypergraph H = ( V, {ei : i ~ 1-}) is isomorphic to the fundamental hypergraph associated with a base of a binary matroid M. The matroid M is uniquely determined by H up to isomorphism.
2. Characterization Binary simpliciat matroid were defined in [1], see also [5]. L e t T be a finite set and 0 < k < [T[ an integer. W e denote by Pk(T) the set of all k-subsets of T. A set C ~ Pk(T) is a cycle iff any (k - 1)-subset of T is contained in an even n u m b e r of k-sets of C. T h e set c~ of all minimal n o n - e m p t y cycles is the set of circuits of the 0012-365X/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
106
.I. T~ma
binary simplicial matroid S k ( T ) = (Pk(T), qg). T h e sets Pk(A) for A ~ Pk+I(T) are the simplest examples of circuits. Lernma 1 can be used to transform certain problems on binary matroids to problems on hypergraphs. W e describe a special base of Sk(T) and use the result of [4] to obtain a characterization of binary simplicial matroids. A similar characterization was given by M.J. T o d d in [3]. O u r conditions are m o r e local, the only global property we need is non-separability of Sk(T). For a finite set T and a non-negative integer k < ITI we define the simplicial hypergraph Hk(T) to be the hypergraph
Hk(T) = (Pk (T), {Vk(A): A ~ Pk+~(T)}). Lemmua 2. The binary simplicial matroid Sk(T) has a base with the associated fundamental hypergraph isomorphic to the simplicial hypergraph H k _ t ( T - { i } ) , for i~T. l~toot. T a k e an arbitrary i e T. T h e set Tk(i) of all k-subsets of T containing i is clearly a base of Sk(T), and for any A e P k ( T ) - T k ( i ) , the set P k ( A U{i}) is the fundamental circuit associated with Tk(i) and A. Now the m a p p i n g D ~ D \ { i } is the desired isomorphism of the fundamental hypergraph associated with Tk(i) and the simplicial hypergraph Hk_~(T-{i}). T o state our characterization of binary simplicial matroids we need some preliminary notions. Let H = (V, {el: i e / } ) be a hypergraph. A vertex v is adjacent to a vertex w itt there is an edge e~ containing both of them. A vertex v is adjacent to an edge e~ iff it is not on e~ but it is adjacent to at least one vertex of el. T h e hypergraph H is connected if[ for any two vertices v, w there is a sequence v = Vo, Vl . . . . . v,, = w such that vi is adjacent to V~+l for i = 0, 1 . . . . . m - 1. (We never assume that v and w are distinct, hence a hypergraph with o n e vertex and without n o n - e m p t y edges is not connected.)
Theorem. A binary matroid M is isomorphic to the binary simplicial matroid Sk (T) for a set T i f f it has a base B such that the fundamental hypergraph H associated with B satisfies the following conditions: (1) H is connected and contains no empty edges. (2) H contains an edge of cardinality k. (3) A n y two edges of H have at most one vertex in common. (4) If a vertex v is adjacent to an edge e, then it is adjacent to exactly two vertices ore. (5) I f vl, v2, v3, v4 are distinct vertices and ex, e2, ea, e4 are distinct edges such that vl ~ ei for i = 1 . . . . . 4, vi ~ ei+l for i = 1, 2, 3, v4 ~ el, and el f) e3 = (~ = e2 tq e4, then Vl is adjacent to v3 and v2 is adjacent to v4. Proof. By L e m m a s 1 and 2, a binary matroid M is isomorphic to the binary
Note on bina13, simplicial matroids
107
simplicial matroid Sk(T) for s o m e T iff it has a base with the associated fundamental hypergraph isomorphic to the simplicial hypergraph Hk_l(T\{i}) for i ~ T. N o w for k > 1 the t h e o r e m follows from the main result of [4]. The case k = 1 is trivial.
References [1] H.H. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory II, Combinatorial Geometries (M.I.T. Press, Cambridge, MA, 1970). [2] A. Schrijver, Matroids and linking systems, J. Combin. Theory (B) 26 (1979) 349-369. [3] M.J. Todd, Characterizing binary simplicial matroids, Discrete Math. 16 (1976) 61-70. [4] J. Tfama, A structure theorem for lattices of generalized partitions, in: Lattice Theory, Colloq. Math. Soc. J~nos Bolyal (North-Holland, Amsterdam, 1980). [5] D.J.A. Welsh, Matroid Theory (Academic Press, New York, 1976).
105
NOTE
NOTE ON BINARY SIMPLICIAL MATROIDS
Mathematical Institute, Czechoslovak Academy of Sciences, ~itnd 25, 115 67 Prague 1, Czechoslovakia Received 13 January 1983 Using an earlier characterization of simplicial hypergraphs we obtain a characterization of binary simplicial matroids in terms of the existence of a special base.
1. Introduction W e shall assume basic knowledge of matroid theory. For undefined notions, see [5]. Let B be a base of a matroid M =" (E, qg), where ~ denotes the set of circuits of M. For any a ~ E \ B there is a unique circuit Ca - B tA{a}. T h e circuit Ca is called the fundamental circuit associated with B and a. A matroid M is binary iff, for any base B of M, every circuit of M is a symmetric difference of fundamental circuits associated with B. H e n c e any binary matroid is uniquely determined by the set of fundamental circuits associated with any base. For a binary matroid M = (E, q~) and a base B of M we define the fundamental hypergraph associated with B to be the hypergraph (B,{Caf'IB: a c E \ B } ) . T h e fundamental hypergraph can have e m p t y edges (corresponding to loops of M), o n e - e l e m e n t and multiple edges (both corresponding to parallel elements of M). Since any matrix o v e r GF(2) is the incidence matrix of a hypergraph, we have the following
Lemmua 1. A n y hypergraph H = ( V, {ei : i ~ 1-}) is isomorphic to the fundamental hypergraph associated with a base of a binary matroid M. The matroid M is uniquely determined by H up to isomorphism.
2. Characterization Binary simpliciat matroid were defined in [1], see also [5]. L e t T be a finite set and 0 < k < [T[ an integer. W e denote by Pk(T) the set of all k-subsets of T. A set C ~ Pk(T) is a cycle iff any (k - 1)-subset of T is contained in an even n u m b e r of k-sets of C. T h e set c~ of all minimal n o n - e m p t y cycles is the set of circuits of the 0012-365X/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
106
.I. T~ma
binary simplicial matroid S k ( T ) = (Pk(T), qg). T h e sets Pk(A) for A ~ Pk+I(T) are the simplest examples of circuits. Lernma 1 can be used to transform certain problems on binary matroids to problems on hypergraphs. W e describe a special base of Sk(T) and use the result of [4] to obtain a characterization of binary simplicial matroids. A similar characterization was given by M.J. T o d d in [3]. O u r conditions are m o r e local, the only global property we need is non-separability of Sk(T). For a finite set T and a non-negative integer k < ITI we define the simplicial hypergraph Hk(T) to be the hypergraph
Hk(T) = (Pk (T), {Vk(A): A ~ Pk+~(T)}). Lemmua 2. The binary simplicial matroid Sk(T) has a base with the associated fundamental hypergraph isomorphic to the simplicial hypergraph H k _ t ( T - { i } ) , for i~T. l~toot. T a k e an arbitrary i e T. T h e set Tk(i) of all k-subsets of T containing i is clearly a base of Sk(T), and for any A e P k ( T ) - T k ( i ) , the set P k ( A U{i}) is the fundamental circuit associated with Tk(i) and A. Now the m a p p i n g D ~ D \ { i } is the desired isomorphism of the fundamental hypergraph associated with Tk(i) and the simplicial hypergraph Hk_~(T-{i}). T o state our characterization of binary simplicial matroids we need some preliminary notions. Let H = (V, {el: i e / } ) be a hypergraph. A vertex v is adjacent to a vertex w itt there is an edge e~ containing both of them. A vertex v is adjacent to an edge e~ iff it is not on e~ but it is adjacent to at least one vertex of el. T h e hypergraph H is connected if[ for any two vertices v, w there is a sequence v = Vo, Vl . . . . . v,, = w such that vi is adjacent to V~+l for i = 0, 1 . . . . . m - 1. (We never assume that v and w are distinct, hence a hypergraph with o n e vertex and without n o n - e m p t y edges is not connected.)
Theorem. A binary matroid M is isomorphic to the binary simplicial matroid Sk (T) for a set T i f f it has a base B such that the fundamental hypergraph H associated with B satisfies the following conditions: (1) H is connected and contains no empty edges. (2) H contains an edge of cardinality k. (3) A n y two edges of H have at most one vertex in common. (4) If a vertex v is adjacent to an edge e, then it is adjacent to exactly two vertices ore. (5) I f vl, v2, v3, v4 are distinct vertices and ex, e2, ea, e4 are distinct edges such that vl ~ ei for i = 1 . . . . . 4, vi ~ ei+l for i = 1, 2, 3, v4 ~ el, and el f) e3 = (~ = e2 tq e4, then Vl is adjacent to v3 and v2 is adjacent to v4. Proof. By L e m m a s 1 and 2, a binary matroid M is isomorphic to the binary
Note on bina13, simplicial matroids
107
simplicial matroid Sk(T) for s o m e T iff it has a base with the associated fundamental hypergraph isomorphic to the simplicial hypergraph Hk_l(T\{i}) for i ~ T. N o w for k > 1 the t h e o r e m follows from the main result of [4]. The case k = 1 is trivial.
References [1] H.H. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory II, Combinatorial Geometries (M.I.T. Press, Cambridge, MA, 1970). [2] A. Schrijver, Matroids and linking systems, J. Combin. Theory (B) 26 (1979) 349-369. [3] M.J. Todd, Characterizing binary simplicial matroids, Discrete Math. 16 (1976) 61-70. [4] J. Tfama, A structure theorem for lattices of generalized partitions, in: Lattice Theory, Colloq. Math. Soc. J~nos Bolyal (North-Holland, Amsterdam, 1980). [5] D.J.A. Welsh, Matroid Theory (Academic Press, New York, 1976).