On the activities of p -basis of matroid perspectives

Discrete Mathematics 339 (2016) 1629–1639

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On the activities of p-basis of matroid perspectives Koko K. Kayibi a , S. Pirzada b,∗ a

Department of Mathematics and Physics, University of Hull, Yorkshire, UK

b

Department of Mathematics, University of Kashmir, Srinagar, India

article

info

Article history: Received 27 March 2015 Received in revised form 17 January 2016 Accepted 21 January 2016 Available online 17 February 2016 In memory of Michel Las Vergnas Keywords: Matroid perspective p-basis Tutte polynomial

abstract There is a renewed interest in matroid perspectives, either for their relevance in other fields of combinatorics and topology, or their applications in engineering. But, like for most of the Tutte invariants, computing the Tutte polynomial of matroid perspectives is #P-hard. Hence the importance of results whose applications would help to speed up computations. In the present paper, we show that a pseudobasis of a matroid perspective can be decomposed by a cyclic flat into two subsets, one of which has zero internal activity and the other has zero external activity. Apart from its own interest in understanding the internal structures of matroid perspective, this decomposition allows an expansion of the Tutte polynomial of matroid perspective over cyclic flats. This can be used to speed up the computation of various evaluations of the polynomial. © 2016 Elsevier B.V. All rights reserved.

1. Introduction A matroid is an independence system satisfying Steinitz exchange lemma. Paradigmatic examples of matroids are defined on the edge set of a graph or the columns of a matrix. Indeed, if G is a graph with edge set E, the cycle matroid of G is the matroid whose ground set is E and a subset A ⊆ E is independent if it contains no (simple) cycles. This is indeed an independence system and Steinitz exchange lemma is embodied in the fact that if A and B are independent and |B| > |A|, there is an element e ∈ B such that A ∪ e is independent. For a matrix M, a matroid on the set of columns of M can be defined as follows. A subset A of the columns of M is independent if it is linearly independent. Steinitz exchange lemma follows from basic linear algebra. The Tutte polynomial of a matroid is a two-variable polynomial whose evaluations at different values of the variables embodies many combinatorial properties of the matroid. The notion of a map between the matroids can be thought of as the analogous of continuous maps in metric topology. Indeed, for the matroids M and N with the same underlying set, there is a weak map from the matroid M to the matroid N if every independent set in N is independent in M, whereas a weak map is a strong map if every closed set of N is closed in M. A strong map is a matroid perspective if M and N are defined on the same set. A typical example of matroid perspective is as follows. Let G be a graph with vertex set V and edge set E. A cycle matroid of G is the matroid defined on E and whose circuits are the cycles of G. Let A be a subset of the vertex set V and let G′ be the graph obtained from G by identifying all the vertices in A to a single vertex, as shown in Fig. 1. It is routine to check that every cycle of G is a union of cycles in G′ . Thus, there is a strong map from the cycle matroid of G to the cycle matroid of G′ . Let B be a subset of the set of edges. The subset B is a closed set if every other edge e not in B does not form a cycle with

∗

Corresponding author. E-mail addresses: [email protected] (K.K. Kayibi), [email protected] (S. Pirzada).

http://dx.doi.org/10.1016/j.disc.2016.01.013 0012-365X/© 2016 Elsevier B.V. All rights reserved.

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Fig. 1. G′ is obtained from G by identifying vertices 1, 2 and 3.

elements of B. We can check that every closed set of G′ is closed in G. Thus, there is a strong map from the cycle matroid of G to the cycle matroid of G′ . A better exposition to the topic of maps between matroids and many more examples can be found in [13]. An extension of the Tutte polynomial to matroid perspectives is the polynomial T (P ; x, y, z ), defined and studied by Las Vergnas [5,7–11]. One of its most interesting evaluations is T (P ; 0, 0, 1). An oriented matroid is a matroid where an orientation is assigned to every element e. One of the simplest examples of oriented matroids is the cycle matroid of a graph G whose edges are oriented. If P is a perspective from an oriented matroid M to the oriented matroid N, then T (P ; 0, 0, 1) counts the number of subsets A such that A is acyclic in M and totally cyclic in N [8]. An obvious application is when there is a strong map from a cycle matroid of a graph G to a cycle matroid of a graph G′ and we define the orientation on the edges of G. This orientation is carried to the edges of G′ in an obvious way. Then, T (P ; 0, 0, 1) counts the number of subsets A of edges, such that A is acyclic in G and totally cyclic in G′ . This evaluation is paramount as it generalizes results on bounded regions of real hyperplane arrangements [15], non Radon partitions of real spaces [3]. More of such applications can be found in [11]. Moreover, the bond matroid of a graph G is the matroid whose independent sets are the subsets of edges of G that do not contain cutsets. Suppose that G and G∗ are dually embedded on a surface. Then there is a matroid perspective from the bond matroid of G∗ to the cycle matroid of G. For the case of 4-valent graphs embedded in the projective plane or a torus, Las Vergnas in [10] relates the Tutte polynomial of matroid perspective to Eulerian tours and cycle decompositions of G. This result sparks a renewed interest in the Tutte polynomial of matroid perspective because of its connection with the Bollobas–Riordan polynomial and Kruskal polynomial, which find many applications in the theory of graphs embedding on surfaces [4]. A generalization of matroid perspectives to a sequence of perspectives in [1] finds applications in electrical network theory. More applications of strong maps in engineering and in the theory of rigidity matroids can be seen in [2,13]. Although the Tutte polynomial and its extensions to other combinatorial structures have many useful applications, evaluating the Tutte polynomial of a matroid is #P. So, a fortiori, is the evaluation of any extension of the Tutte polynomial, such as the Tutte polynomial of a perspective. Hence any little help to speed up the computation is welcome. Indeed, computing the Tutte polynomial by deletion/contraction recursion entails a number of steps that is exponential in n, where n is the number of elements in the matroid M. Suppose that M is decomposed in to two parts M1 and M2 of cardinality n1 and n2 , respectively. If we evaluate the Tutte polynomial of a matroid M as a product T (M ; x, y) = T (M1 ; a, b)T (M2 ; c , d), the computation will be speeded up by a factor of 2 present paper.

n−n

1

, where n1 ≥ n2 . This justifies the relevance of Theorem 4 of the

2. Definitions The unfolding of the present paper will require the following definitions and the known results. A matroid M defined on a finite nonempty set E consists of the set E and a collection C of subsets of E, satisfying the following axioms: C1: ∅ ̸∈ C C2: if C1 and C2 are in C , and C2 ⊆ C1 , then C2 = C1 . C3: if C1 and C2 are in C , e ∈ C1 ∩ C2 and f ∈ C1 \ C2 , then there is a C3 such that f ∈ C3 and C3 ⊆ (C1 ∪ C2 ) \ e. Elements of C are the circuits of the matroid M. Axiom C3 is the strong circuit elimination axiom. A subset I ⊆ E is independent in M if I contains no circuit. A subset B ⊆ E is a basis of M if B is a maximal independent set of M. The rank function of M, denoted by rM , is an integer valued function such that for A ⊆ E, rM (A) is the cardinality of the largest independent set contained in A. We write r instead of rM if the context is obvious. A subset F is a flat of M, if for all e ̸∈ F we have r (F ∪ e) = r (F ) + 1. A flat of M is a cyclic flat if it is the union of circuits of M. For a subset X ⊆ E, the closure of X, denoted by cl(X ), is the smallest flat containing X . Definitions of matroids using the independence, rank, basis or closure axioms can be seen in [12]. For X ⊆ E and A ⊆ X , the matroid M |X , the restriction to X , is the matroid defined on X with a rank function rM |X (A) = rM (A). For X ⊆ E and A ⊆ E \ X , the matroid M /X , the contraction by X , is the matroid defined on E \ X

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with a rank function rM /X (A) = rM (A ∪ X ) − rM (X ).

(1)

Suppose that E is totally ordered and let B be a basis of M. An element e ∈ B is internally active w.r.t. B if e is the least ∗ element of the (unique) cocircuit C ⊆ (E \B) ∪ e. An element e ̸∈ B is said to be externally active w.r.t. B if e is the least element of the (unique) circuit C ⊆ B ∪ e. The above unique cocircuit (respectively circuit) is called the fundamental cocircuit (respectively fundamental circuit) of the element e. We denote by eM (B) (respectively iM (B)) the number of elements of E which are externally (respectively internally) active w.r.t. B. We may simply write e(B) or i(B) when the context is obvious. An internal basis (external basis) is a basis B such that e(B) = 0 (i(B) = 0). The following result, proved in [5], provides a relation between the cyclic flats and the activities of the basis of matroids. Theorem 1 ([5]). Let M be a matroid on a linearly ordered set E and B be a basis of M. There exists a unique cyclic flat F such that B ∩ F is an external basis of M (F ) and B \ F is an internal basis of M /F . Moreover, eM (B) = eM |F (B ∩ F ) and iM (B) = iM /F (B \ F ). The main result of this paper is an extension of Theorem 1 for matroid perspectives (see Theorem 2 in Section 3). Let M and N be two finite matroids defined on the same underlying set E with rank functions r and s, respectively. We call the pair P = (M , N ) a matroid pair. There is a weak map from M to N if every independent set of N is independent in M, whereas there is a strong map from M to N if every flat of N is a flat of M. If there is a strong map from M to N, the pair P = (M , N ) is called a matroid perspective. In a matroid perspective (M , N ), the following are equivalent. (i) Every circuit of M is a union of circuits of N. (ii) Every flat of N is a flat of M. (iii) clM (X ) ⊆ clN (X ) for every X ⊆ E. For X ⊆ E, we shall write P \X for (M \X , N \X ), and P /X for (M /X , N /X ). The notation P |X stands for (M |X , N |X ), where for any matroid M, M |X denotes the restriction of the matroid M to the subset X . A flat of P is a subset F ⊆ E such that for all e ̸∈ F r (F ∪ e) + s(F ∪ e) > r (F ) + s(F ), that is, a flat of P is a subset that is a flat (in the matroid sense) of M. A flat of P which is a union of circuits in N is called a cyclic flat of P. In other words, a cyclic flat of P is a set F such that F is a flat of P and P |F contains no coloop in N. A pseudobasis (or just p-basis) of P is a subset B ⊆ E such that B is independent in M and spanning in N. If E is totally ordered, following [9], we say that an element e ∈ B is internally active with respect ∗ ∗ to B if there exists a (unique) cocircuit C ⊆ (E \B) ∪ e in N, and e is the least element of C . An element e ̸∈ B is said to be externally active with respect to B if there exists a (unique) circuit C ⊆ B ∪ e in M, and e is the least element of C . The above unique cocircuit (respectively circuit) is called the fundamental cocircuit (respectively fundamental circuit) of the element e. We denote by e(B) (respectively i(B)) the number of elements of E which are externally (respectively internally) active with respect to the p-basis B. 3. Decomposition of p-bases of a matroid perspective The following result shows that a p-basis of a matroid perspective can be decomposed by a cyclic flat into two subsets, one of which has zero internal activity and the other has zero external activity. This result generalizes Theorem 5.1 in Etienne and Las Vergnas [5]. Theorem 2. Let P be a matroid perspective defined on a totally ordered set E. Then for every p-basis B of P, there is a cyclic flat F of P such that (i) B1 = B ∩ F is a p-basis of P |F of zero internal activity. (ii) B2 = B\F is a p-basis of P /F of zero external activity. (iii) e(B) = e(B1 ) and i(B) = i(B2 ). (iv) Furthermore, for every cyclic flat F , there is a p-basis B of P such that B1 = B ∩ F and B2 = B\F , and that satisfies conditions (i)–(iii). Before we give the proof of Theorem 2, we have some construction algorithms and lemmas. The following algorithm constructs F , B1 and B2 from B. Algorithm 1. Input: a p-basis B of a matroid perspective P defined on a totally ordered set E. Outputs: B1 , B2 such that B = B1 ∪ B2 , and a cyclic flat F such that B1 = B ∩ F and B2 = B \ F . Step 1: Set B1 = B and B2 = ∅. Step 2: Let F be the smallest flat of P containing B1 . Step 3: If B1 contains an element e which is internally active w.r.t. B1 , considered as a p-basis of F , then let B1 = B1 \e and B2 = B2 ∪ e, and go back to step 2. If no such element exists, then output B1 , B2 and F .

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First, we notice that Algorithm 1 is well defined, in the sense that the output does not depend on the particular choice of the internally active element at each step. Proposition 1. Let the two elements e and e′ are both internally active w.r.t. B1 , considered as a p-basis of a flat K . Let F and F ′ be the smallest flats containing B1 − e and B1 − e′ , respectively. Then e is internally active w.r.t. B1 − e′ considered as a p-basis of F ′ if and only if e′ is internally active w.r.t. B1 − e considered as a p-basis of F . (This essentially means that if an element is internally active at the beginning of Algorithm 1, it stays internally active until it is removed.) Proof. An element e is internally active w.r.t. B1 , considered as a p-basis of K , if it is the least element of its fundamental cocircuit in N |K . Since e and e′ are both internally active w.r.t. B1 , their fundamental cocircuits in N |K are distinct (disjoint ∗ ∗ or not) cocircuits of N |K . Let C and D denote the fundamental cocircuits of e and e′ , respectively. Suppose that Algorithm 1 picks e. Now, the smallest flat F containing B1 − e, is obtained from K by removing all the elements that belong to the ∗ ∗ ∗ ∗ ∗ fundamental cocircuit of e. That is, F = K − C . Since C and D are distinct, it is routine to check that D − C is not empty ∗ ∗ ′ (since it must contain at least e ) and is a cocircuit of N |F . Obviously, D − C is the fundamental cocircuit of e′ in N |F and e′ is still the least element of it. Thus e′ is internally active w.r.t. B1 − e considered as a p-basis of F .  Lemma 1. Algorithm 1 terminates and returns a cyclic flat F . Proof. Let us assume to the contrary that Algorithm 1 does not terminate. This is possible if either (a) the loop going back to Step 2 does not end, or (b) there exists a set B1 such that clM (B1 ) does not exist. Case (a) is not possible, since P is a pair of finite matroids and (b) contradicts the definition of clM (B1 ), which always exists. It remains to show that the set F returned by Algorithm 1 is a cyclic flat. Assume that F is not cyclic. Then N |F contains a coloop e. So e must be an element of B1 (since every other element of F is in clM (B1 ) ⊆ clN (B1 )). Therefore there is a unique cocircuit of N |F contained in (F \B1 ) ∪ e, and this cocircuit is equal to {e}. So, e is internally active with respect to B1 , considered as a p-basis of N |F . This contradicts the construction of F . Therefore the flat F is a cyclic flat.  Lemma 2. The set B2 output by Algorithm 1 has zero external activity. Proof. Let B = {e1 , . . . , er } and assume that when the algorithm terminates, B2 = {e1 , . . . , ek }, while B1 = {ek+1 , . . . , er }. Define F0 to be E, and for i = 1, . . . , k, let Fi = clM (B − {e1 , . . . , ei }). Assume that ei is internally active in P |Fi−1 with respect to the p-basis {e1 , . . . , ei }. Therefore there is a cocircuit Xi∗ of N |Fi−1 such that ei ∈ Xi∗ , and Xi∗ ⊆ (Fi−1 − B) ∪ ei , and ei is the least element of Xi∗ . If i > 1, then Xi∗ is a cocircuit in N |Fi−1 , but it may not be a cocircuit of N. It is contained in a cocircuit Yi∗ of N, where ∗ Yi ∩ Fi−1 = Xi∗ and Yi∗ is contained in E − Fi ⊆ E − Fk . (Define Y1∗ to be X1∗ .) Note that ei may not be the least element of Yi∗ . ∗

∗

If N is a strong-map image of M, then M is a strong-map image of N . (See [12], Propositions 7.3.1 and 7.3.6.) Therefore every cocircuit Yi∗ is a union of cocircuit of M. Let Zi∗ be a cocircuit of M such that ei ∈ Zi∗ ⊆ Yi∗ . Assume that x ∈ E − (Fk ∪ B2 ) is externally active in P /Fk with respect to the p-basis B2 = {e1 , . . . , ek }. Then there is a circuit C of M /Fk such that x ∈ C ⊆ B2 ∪ x and x is the least element of C . Note that C ̸= {x}, since x would be a loop of M /Fk , which means that x ∈ clM (Fk ), and therefore x ∈ Fk , as Fk is a flat of M. Therefore, C ∩ B2 ̸= φ . Let t be the smallest index so that et ∈ C . Then x is not in Xt∗ , since x is the smallest element in C and et is the smallest element in Xt∗ . But C is a circuit of M /Fk , and Zt∗ is a cocircuit of M and therefore a cocircuit of M /Fk . They both contain et , and they cannot intersect in a single element. The only other element that can be in their intersection is x, as C ⊆ B2 ∪ x and Zt∗ ⊆ (E − (Fk ∪ B2 )) ∪ et . Therefore x ∈ Zt∗ − Xt∗ . Since Z1∗ ⊆ Y1∗ = X1∗ , this means that t > 1. Note that x is in Yt∗ ⊆ Zt∗ , but x is not in Ft −1 , as Yt∗ ∩ Ft −1 = Xt∗ and x ̸∈ Xt∗ . Since C is contained in {et , et +1 , . . . , ek } ∪ {x}, it is contained in Ft −1 , apart from the element x. This implies that x ∈ clM /F (Ft −1 − Fk ), so x ∈ clM (Ft −1 ). But clM (Ft −1 ) = Ft −1 , k

since Ft −1 is a flat of M. Therefore x is in Ft −1 and we have a contradiction.



Lemma 3. e(B) = e(B1 ) and i(B) = i(B2 ). Proof. First, we prove that e(B) ⊆ e(B1 ). Let B be a p-basis of (M , N ), and e ̸∈ B be externally active w.r.t. B. Thus there is a circuit C ⊆ B ∪ e of M, and e is the least element in C . If f ∈ C ∩ B, then (E \B) ∪ f either contains cocircuit in N, or it does not. ∗ Assume that (E \B) ∪ f contains a cocircuit C of N. Then B − f is contained in a hyperplane, and therefore is not spanning ∗ in N. As B is spanning in N, it follows that rN (B − f ) = r (N ) − 1, so clN (B − f ) is a hyperplane, and C = E − clN (B − f ). The fact that B is spanning in N, but B − f is not, means that f ̸∈ clN (B − f ). Thus there is no circuit of N contained in B that contains f . However, C ⊆ B ∪ e is a circuit of M, and f ∈ C . Since C is a union of circuits in N, there is a circuit C ′ ⊆ C that ∗ contains f . By the previous argument, C ′ in not contained in B, so e ∈ C ′ . Now assume that e ̸∈ C . Then e is in the closure ′′ ′′ ′′ of B − f in N. This means that there is a circuit C of N such that e ∈ C and C ⊆ (B − f ) ∪ e. Now C ′ and C ′′ are distinct circuits of N that both contain e, and f is in C ′ − C ′′ . By applying the strong circuit-exchange axiom, we see that there is a circuit contained in (C ′ ∪ C ′′ ) − e that contains f . But this circuit is contained in B, which is a contradiction. This shows that ∗ ∗ e ∈ C . So f is not the least element of C and therefore it belongs to B1 , by construction. This implies that the element e is externally active w.r.t. B1 . Hence, e(B) ⊆ e(B1 ).

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It is routine to reverse the implications in the above argument to prove that e(B1 ) ⊆ e(B). It is also routine to use a dual argument to prove that i(B) = i(B2 ).  We recall that Algorithm 1 inputs a p-basis B and returns B1 and B2 and a cyclic flat F that satisfy properties (i)–(iii). Part (iv) of Theorem 2 asserts that, given a cyclic flat F , there exists a p-basis B = B1 ∪ B2 , where B1 = B ∩ F is a p-basis of P |F of zero internal activity, and B2 = B \ F is a p-basis of P /F of zero external activity. Algorithm 2 shows that B1 can be constructed from F . Algorithm 2. Input: a cyclic flat F . Output: B1 such that B1 contains no element that is the least element of its fundamental cocircuit in N |F . Let P |F = {e1 , . . . , er }, where e1 > e2 > · · · > er in the total ordering of elements of E. Step 1: Set B1 = φ . Step 2: Choose an element e ∈ F \B1 , where e is the largest element such that B1 ∪ {e} does not contain a cycle of M |F . Set B1 = B1 ∪ {e}. Step 3: If B1 ∪ {e} spans N |F , stop. Else, go back to Step 2. Note that Algorithm 2 is just a greedy algorithm for constructing a ‘‘largest possible’’ independent set of M |F that spans N |F . We only have to prove that it terminates and that B1 has zero internal activity. An independent set B of a matroid M is augmented by an element e if B ∪ e is independent. An algorithm is said to stall at its ith step if the conditions required to proceed to the next step cannot be met at the ith step. Lemma 4. Algorithm 2 terminates. Proof. Let t denote the rank of the matroid N |F . If t ≤ 1, then Algorithm 2 terminates, since it only needs to pick the largest element of P |F . So, suppose t > 1 and Algorithm 2 does not terminate. Let B′1 ∪ {e1 , . . . , er } be an independent set of M |F of maximal cardinality, where Algorithm 2 stalls. That is, for all the elements e in F \(B′1 ∪ {e1 , . . . , er }), we have either (a), (B′1 ∪ {e1 , . . . , er }) ∪ e is not independent in M |F or (b), (B′1 ∪ {e1 , . . . , er }) ∪ e does not span N |F . If case (a) occurs, then B′1 ∪ {e1 , . . . , er } is a basis of M |F , and thus it must span N |F . This is a contradiction. Suppose that case (b) occurs. That is, (B′1 ∪ {e1 , . . . , er }) ∪ {e} is independent, but does not span N |F . Then Algorithm 2 does not stall as it can choose another element f in F \(B′1 ∪ {e1 , . . . , er } ∪ {e}), or falls in case (a) or (b) for the element f . Thus, by induction, the algorithm always terminates.  Lemma 5. The set B1 output by Algorithm 2 has zero internal activity. Proof. We are required to show that B1 has no element which is the least element of its fundamental cocircuit in N |F . Suppose, for a contradiction, that the cardinality of B1 is r, and that B1 contains an element e which is the least element of its fundamental cocircuit in M |F . If r = 1, then e1 is necessarily the biggest element of its fundamental cocircuit in M |F , since Algorithm 2 picks the largest element of F without any restriction. Now, assume that e ∈ B1 is the least element of its fundamental cocircuit in N |F , and let C ∗ be that fundamental cocircuit. But, C ∗ is a union of cocircuits of M |F , one of whose is the fundamental cocircuit of e in M |F . Therefore e is the least element of its fundamental cocircuit in M |F . This is a contradiction. For induction, suppose that B1 = {e1 , . . . , er }, where ei is not the least element of its fundamental cocircuit in M |F , for all i from 1 to k, with 1 ≤ k ≤ r. Consider ek+1 and let Ck∗+1 = {f1 , f2 , . . . , ek+1 }, where f1 > f2 > · · · > ek+1 , be the fundamental cocircuit of ek+1 in M |F . Now, Algorithm 2 cannot pick any of the fj in Ck∗+1 only if each such fj forms a circuit Cj in M |F with some elements of the set {e1 , . . . , ek }. But this means that r ({e1 , . . . , ek } ∪ {fj }) = r ({e1 , . . . , ek }) for every such fj . So ek+1 is the unique element of Ck∗+1 in M |F . That is, ek+1 is a coloop in M |F . Since M |F is a cyclic flat, so ek+1 must be an element of some circuit C of N |F . If C is a loop in N |F , then vacuously, ek+1 is not the least element of its fundamental cocircuit in N |F . If |C | > 1, let D∗ be the fundamental cocircuit of ek+1 in N |F . The element ek+1 is the least element of D∗ only if all the other elements of D∗ are greater than ek+1 . Now, D∗ = D∗1 ∪ D∗2 ∪ · · · ∪ D∗s , where all the D∗i ’s are cocircuits of M |F and D∗1 = Ck∗+1 = {ek+1 }. Since all the elements of D∗2 are greater than ek+1 , Algorithm 2 will not pick any of them only if each of them forms a circuit in M |F with some elements of {e1 , . . . , ek }. But then D∗2 is not a cocircuit of M |F . This is a contradiction. Thus the lemma holds by induction.  The following algorithm gives the construction of B2 for a given cyclic flat F . Algorithm 3. Input: a cyclic flat F . Output: B2 such that for all elements e ∈ (P \F )\B2 , the element e is not the least element of its fundamental circuit in M |(P \F ). Let P \F = {e1 , . . . , er }, where e1 > e2 > · · · > er in the total ordering of elements of E. Step 1: Set B2 = φ . Step 2: Pick an element e, where e ∈ (P \F )\B2 and e is the least element such that B2 ∪ {e} does not contain a cycle of M |(P \F ). Set B2 = B2 ∪ {e}. Step 3: If B2 ∪ {e} spans N |(P \F ), stop. Else, go back to Step 2.

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Note that Algorithm 3 is the same greedy algorithm as Algorithm 2, except that Step 2 picks the least available element. Thus, using a similar argument as for Algorithm 2, it is routine to check that Algorithm 3 terminates. Now, we prove that the set B2 output by Algorithm 3 has zero external activity. Lemma 6. The set B2 output by Algorithm 3 has zero external activity. Proof. Assume to the contrary that e is an element of ((P \F )\B2 ) and that e is the least element of its fundamental circuit with respect to B2 in M |(P \F ). Let Ce = {f1 , f2 , . . . , ft , e}, where f1 > f2 > · · · > ft > e be the fundamental circuit of e in M |(P \F ) with respect to B2 . Now, Algorithm 3 will pick fi instead of e only if e will form a circuit with the elements already chosen. That is, the element e will have another fundamental circuit that is different from Ce . This contradicts the uniqueness of fundamental circuits in M |(P \F ).  Lemma 7. B = B1 ∪ B2 is a p-basis of P. Proof. With B1 and B2 as defined above, we first show that r (B1 ∪ B2 ) = r (B1 ) + r (B2 ).

(2)

Indeed, by the semi modularity property of the rank function of a matroid, we have r (B1 ∪ B2 ) + r (B1 ∩ B2 ) ≤ r (B1 ) + r (B2 ), which becomes r (B1 ∪ B2 ) ≤ r (B1 ) + r (B2 ),

(3)

since B1 and B2 are disjoint. Now, suppose that inequality holds in Eq. (3). Then there is a circuit C = (D1 ∪ D2 ) ⊆ B1 ∪ B2 , where D1 ⊆ B1 and D2 ⊆ B2 . By Eq. (1), we have rM /D (D2 ) = r (D2 ∪ D1 ) − r (D1 ). 1

Now, D2 is independent in M /D1 , since D2 ⊆ B2 . So B2 is independent in M /F and M /F ⊆ M /D1 . Moreover, D1 is independent in M |F , since D1 ⊆ B1 . So

|D2 | = r (D2 ∪ D1 ) − |D1 |. Therefore D2 ∪ D1 is independent as r (D2 ∪ D1 ) = |D2 | + |D1 |. This is a contradiction. Also, we need to show that r (B2 ) = rM /F (B2 ).

(4)

Indeed, by Eq. (1) rM /F (B2 ) = r (B2 ∪ F ) − r (F )

= r (B2 ∪ B1 ) − r (B1 ), since B1 is a p-basis of P |F = r (B2 ) + r (B1 ) − r (B1 ), by Eq. (3) which is Eq. (4). Now, r (B) = r (B1 ∪ B2 ) = r (B1 ) + r (B2 ),

by Eq. (3)

= rM |F (B1 ) + rM /F (B2 ), by Eqs. (2) and (4), = |B1 | + |B2 |, since both are independent sets, = |B|, since they are disjoint. Thus B is independent in M. Similarly, s(B) = s(B1 ∪ B2 ) = s(B1 ) + s(B2 )

= sN |F (B1 ) + sN /F (B2 ),

by Eqs. (2) and (5),

= s(N |F ) + s(N /F ), since B1 spans N |F and B2 spans N /F , = s(N ), since they are disjoint. So B is spanning in N. Therefore, B is a p-basis of P.



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Lemma 8. Let F be the input of Algorithm 2 or Algorithm 3, and let B = B1 ∪ B2 , where B1 and B2 are the outputs of Algorithms 2 and 3, respectively. Then Algorithm 1 when applied to B returns F . Proof. Assume that Algorithm 1 is applied to B. Let B2 = {e1 , e2 , . . . , elast } such that e1 < e2 < · · · < elast . Further, let F0 ⊆ F1 ⊆ · · · ⊆ Flast be the sequence of cyclic flats obtained by applying iteratively step 3 of Algorithm 1 with F0 = E and ∗ ∗ ∗ ∗ Fi = cl(B\{e1 , e2 , . . . , ei }). We recall that Fi = E \{e1 , e2 , . . . , ei }\{Ce , Ce , . . . , Ce }, where Ce denotes the fundamental 1

2

i

i

cocircuit of the element ei (as it is at the ith iteration of the algorithm). For some ei ∈ B2 , suppose ei is not internally active with respect to B\{e1 , e2 , . . . , ei }, considered as a p-basis of Fi . Therefore ei is not the least element of its fundamental ∗ ∗ cocircuit Ce ⊆ P \Fi−1 . So there is an element f ∈ Ce and f ̸∈ B\{e1 , e2 , . . . , ei } such that f < ei . But, by assumption, f is i

i

not externally active with respect to B2 . This implies that f is not the least element of its fundamental circuit in B2 . That is, there is an element g ∈ B2 \{e1 , e2 , . . . , ei } such that g < f . Recalling that f is in the fundamental cocircuit of e if and only if e is in the fundamental circuit of f , we get g < ei . This is a contradiction. So the algorithm removes B2 from B. But Flast = F , since F = cl(B1 ). Hence the algorithm outputs F .  We are now prepared for the proof of Theorem 2. Proof of Theorem 2. (i). By construction the subset B1 spans N |F and is independent in M |F , since it is a subset of B. Moreover, by construction, its internal activity is zero. Therefore, it is a p-basis of P |F of zero internal activity. (ii). By Eq. (1), we have rM /F (B2 ) = r (B2 ∪ F ) − r (F ) = r (B) − r (B1 )

= |B| − |B1 | = |B2 |. Also sN /F (B2 ) = s(B2 ∪ F ) − s(F ) = s(B) − s(F ) = sN /F (N /F ). Therefore B2 is a p-basis of P /F , since it is independent in M /F and spanning in N /F . Moreover, by Lemma 2, B2 has zero external activity. (iii). This is proved in Lemma 3. (iv). Let F be a cyclic flat. Define B = B1 ∪ B2 , where B1 is a p-basis of P |F of zero internal activity, and B2 is a p-basis of P /F of zero external activity. Such B1 exists by Algorithm 2, and by Lemma 5, B1 has zero internal activity. By Algorithm 3, B2 exists and by Lemma 6, B2 has zero external activity. Therefore, for the given cyclic flat F , clearly B1 and B2 exist. Moreover, Lemma 7 shows that B = B1 ∪ B2 is a p-basis of P. And, since Lemma 8 asserts that Algorithm 1 when applied to B returns F , we conclude that B1 and B2 have the required properties.  4. Tutte polynomial of matroid perspective over cyclic flats Let M be a matroid defined on a set E. The Tutte polynomial of M is a 2-variable polynomial defined as follows. T (M ; x, y) =

 r (E )−r (X ) |X |−r (X ) (x − 1) (y − 1) , X ⊆E

where r is the rank function of M. Equivalently, the Tutte polynomial can be expanded as T (M ; x, y) =



Tij xi yj .

ij

Theorem 3 ([6]). The Tutte polynomial T (M ; x, y) satisfies T (M ; x, y) =



T (M |A; 0, y)T (M /A; x, 0).

A⊆E

The Tutte polynomial of matroid perspectives, denoted by t (P ; α, β, γ ), is defined for any matroid perspective P on E as follows [7,9,11]. t (P ; α, β, γ ) =

 s(E )−s(X ) |X |−r (X ) r (E )−s(E )−r (X )+s(X ) (α − 1) (β − 1) γ . X ⊆E

In [9,11], it is proved that the Tutte polynomial of matroid perspectives can be expanded as a sum over its p-bases B as follows. t (P ; α, β, γ ) =



α

i(B)

β

e(B)

γ

r (M )−|B|

.

B

Using this, we obtain the following generalization of Theorem 3 in [6].

(5)

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Theorem 4. If P is a matroid perspective defined on a set E, then t (P ; α, β, γ ) =



t (P |F ; 0, β, −1)t (P /F ; α, 0, γ ),

(6)

F ⊆E

where the sum is over cyclic flats of P. We can prove Theorem 4 by checking that Eq. (6) satisfies the deletion/contraction recursion of Tutte invariants. Here we prefer to give a combinatorial proof with the help of Theorem 2. This combinatorial proof of Theorem 4 requires the following definitions and lemmas.  Let T = t ( P | F ; 0, β, −1)t (P /F ; α, 0, γ ). That is, F T =

i(B ) 1

  F

β

0

e(B ) 1

r (P |F )−|B | 1



(−1)

B ∈B(P |F ) 1

α

i(B ) 2

e(B ) 2

0

γ

r (P /F )−|B | 2

,

(7)

B ∈B(P /F ) 2

where B(X ) denotes the set of p-bases of X . Suppose that we expand Eq. (8) (by distributivity) for a given cyclic flat F . Then the monomial corresponding to the pair (B1 , B2 ), such that B1 is a p-basis of P |F and B2 is a p-basis of P /F , is non zero if and only if i(B1 ) = e(B2 ) = 0. That is, if and only if the pair (B1 , B2 ) is such that B1 is a p-basis of P |F of internal activity 0, and B2 is a p-basis of P /F of external activity 0. (We note that the pair (B1 , B2 ), such that B1 ∪ B2 = B and Algorithm 1 applied to B yields F , is only one of the possibly many such pairs.) Using the above remark, Eq. (7) becomes T =



=



F

F

0

0 β

e(B ) 1

(−1)

r (P |F )−|B | 1

α

i(B ) 2

0

0 γ

r (P /F )−|B | 2

B′

β

e(B ) 1

(−1)

r (P |F )−|B | 1

α

i(B ) 2

γ

r (P /F )−|B | 2

,

B′

where B′ denotes the set of p-bases B of F such that B = B1 ∪ B2 , B1 is a p-basis of P |F of internal activity 0 and B2 is a p-basis of P /F of external activity 0. Now, using the facts that e(B1 ) = e(B), i(B2 ) = i(B) and r (P /F ) − |B2 | = r (P ) − |B|, we get T =

 F

β

e(B)

α

i(B)

γ

r (P )−|B|

r (P |F )−|B | 1

(−1)

.

(8)

B(F )

In the matroid pair P |F , the nullity of a set B is the integer r (P |F ) − |B|. In the matroid M, an independent set B is augmentable if B does not span M. An augmentation of B is a set B ∪ e such that B ∪ e is independent. An augmentation pair is the pair {B, B ∪ e} such that B ∪ e is an augmentation of B. The contribution of the set B with respect to the cyclic flat 0

e(B ) 1

r (P |F )−|B | 1

i(B ) 2

0

r (P /F )−|B | 2

F is the monomial xy, where x = 0 β (−1) and y = α 0 γ , and where B1 = B ∩ F is a p-basis of P |F of zero internal activity and B2 = B\F is a p-basis of P /F of zero external activity. Since xy may be negative, according to whether r (P |F ) − |B1 | is odd or even, the main concern of the proof is to show that all the negative terms will be canceled out. First, if we let M and B to denote respectively the set of non-zero monomials of T that are not canceled out and the set of p-basis of P, we get the following fact. Lemma 9. A monomial xy is in M if and only if it is the contribution of a p-basis B = B1 ∪ B2 with respect to the cyclic flat F , such that the following conditions are satisfied. (i) B1 does not belong to an augmentation pair in M |F . (ii) The cyclic flat F is such that B1 = B ∩ F is a p-basis of P |F of zero internal activity and B2 = B\F is a p-basis of P /F of zero external activity. Proof. We first prove the necessity of the two conditions. Suppose that (i) or (ii) (or both) do not hold. Condition (i) implies that there is no subset B′1 such that B1 ⊆ B′1 and whose contribution will cancel out the contribution of B1 . Now, suppose that B1 is augmentable in M |F . Then there is a cocircuit C ∗ of M |F such that B1 does not intersect C ∗ . Consider B′1 = B1 ∪ f , where f is the least of the elements of C ∗ . If the nullity of B1 is p, then the nullity of B1 ∪ f is p − 1. Let B be a basis of a matroid and f ∈ B and g ̸∈ B. Then f belongs to the fundamental circuit of g w.r.t. B if and only if g belongs to the fundamental cocircuit of f w.r.t. B. Using this fact, we get e(B1 ) = e(B′1 ), since f is the least element of all the fundamental circuits of elements that belong to C ∗ . Moreover, since B1 spans N |F , B1 intersects all the cocircuits of N |F . Thus there is no cocircuit containing only f and the elements of F \B1 . So 0 = i(B1 ) = i(B′1 ). Therefore the contribution of B1 cancels out the contribution of B′1 , since they are like terms of opposite signs. If A(F ) denotes the set of all B1 that are augmentable in M |F , and Im(A(F )) denotes the set of augmentations of elements of A(F ), we show that there is a one-one correspondence between A(F ) and Im(A(F )). Indeed, it may be the case that two augmentable sets are matched to the same augmentation, leaving one of them uncanceled. So, let ψ(X ) = X ∪ e, where X ∈ A(F ) and X ∪ e ∈ Im(A(F )). This is a well defined function. We have to show that ψ is injective and surjective. It is obviously surjective, since, given any set X ∪ e, ψ matches it onto X . We only have to show the injection. So, suppose there are two different augmentable p-basis B1 and D1 such that B1 ∪ f = D1 ∪ g. Now, B1 ∪ f = D1 ∪ g

K.K. Kayibi, S. Pirzada / Discrete Mathematics 339 (2016) 1629–1639

1637

Fig. 2. The matroid perspective P = (M , N ).

only if B1 = {g } and D1 = {f }. That is, F = {f , g }, where both f and g are coloops in M |F and loops in N /F . But then the contribution of {f } w.r.t. F will cancel the contribution of ∅ w.r.t. F , leaving the contribution of {g } w.r.t. F to be canceled by that of {f , g }. Therefore, every monomial that is not the contribution of a set satisfying (i) is canceled out. Obviously, if Condition (ii) does not hold, the contribution of B w.r.t. F is zero. To prove the sufficiency, we suppose that (i) and (ii) hold for the p-basis B = B1 ∪ B2 . By (ii), its contribution w.r.t. F is non-null, and cannot be canceled, since B1 has no augmentation in F . Moreover, since B1 is independent and spanning in F , we have that (−1)r (M |F )−|B1 = (−1)0 = 1. Therefore its contribution is positive. Suppose there is another p-basis D, satisfying conditions (i) and (ii) with respect to another flat F ′ and whose contribution w.r.t. F ′ cancel out the contribution of B w.r.t. F . But since the contribution of D w.r.t. F ′ is also positive, the cancellation is impossible. Therefore if B satisfies (i) and (ii) w.r.t. F , its contribution is in M.  Example. Consider the matroid perspective given in Fig. 2, where the total ordering is a > b > c > d. Below are all the p-basis of P with their monomials, as given in Eq. (6). In what follows, we include variables α , β and γ 0

with exponent zero in our monomials, for example, writing α instead of 1. p-bases

monomial

{ a, b } {a, c } {a, d} { b, c } { b, d } {a, b, d} {a, c , d} { b, c , d }

α β γ 0 0 1 α β γ 1 0 1 α β γ 1 0 1 α β γ 2 0 1 α β γ 0 1 0 α β γ 0 0 0 α β γ 1 0 0 α β γ

0

1

1

Below is the contribution of each cyclic flat F , as given in Eq. (7). The column φ(xy √) gives the p-basis of P to which corresponds the monomial xy of the given row. In the column ‘‘Remark’’, the symbol in a row i means that the row i contributes a non null monomial to Eq. (7). The symbol × means that the row contributes zero, since it fails Condition (ii) of Lemma 9. Two rows bear the same number in the column ‘‘Remark’’ if they form an augmentation pair and thus cancel each other, as they fail Condition (i) of Lemma 9. For example, the contribution of the set {a, b} with respect to the flat {a, b, c , d} is canceled out by the contribution of the set {a, b, d}, since the later is an augmentation of the former. But, if considered as a p-basis of the flat {a, b, c }, the set {a, b} obeys Conditions (i) and (ii) of Lemma 9. Hence its contribution is non-zero. F =∅ p-basis

x

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ

0

0

0

P /F = {a, b, c , d} p-bases

y

{a, b} { a, c } { a, d } {b, c } {b, d} {a, b, d} {a, c , d} {b, c , d}

α β γ 0 0 1 α β γ 1 0 1 α β γ 1 0 1 α β γ 2 0 1 α β γ 0 1 0 α β γ 0 0 0 α β γ 1 0 0 α β γ

0

1

Remark 1

× √ √ √ √ × √ √

φ(xy) × {a, c } {a, d} { b, c } { b, d } × {a, c , d} {b, c , d}

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F = {a, b, c , d} p-basis

x

{a, b} { a, c } {a, d} { b, c } {b, d} { a, b , d } {a, c , d} { b, c , d }

α β γ 0 0 1 α β γ 1 0 1 α β γ 1 0 1 α β γ 1 0 1 α β γ 0 1 0 α β γ 0 0 0 α β γ 1 0 0 α β γ

0

F = {a, b, c } p-basis

1

{ a, b } {a, c } { b, c } { a, b } {a, c } { b, c }

1

x

{d } {c } {d , c } {d } {c } {d , c }

α β γ 0 0 1 α β γ 0 0 0 α β γ 1 0 1 α β γ 0 0 1 α β γ 0 0 0 α β γ

1

0

α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ

0

P /F = {d} p-bases 0

α β γ 1 0 0 α β γ 1 0 0 α β γ 0 1 0 α β γ 1 0 0 α β γ 1 0 0 α β γ

F = {d, c } p-basis

y

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

1

x 0

P /F = ∅ p-bases

∅ ∅ ∅ {d} {d} {d}

Remark

0

0

y 0

2

× × × 1 2

× Remark

0

√

1

α β γ 0 0 1 α β γ 0 0 1 α β γ 0 0 0 α β γ 0 0 0 α β γ 0 0 0 α β γ

P /F = {a, b} p-bases

y

{ a} { a} { a} {b} {b} {b}

α β γ 0 1 0 α β γ 0 1 0 α β γ 1 0 0 α β γ 1 0 0 α β γ 1 0 0 α β γ

1

1

0

× × √ × × Remark

1

0

× × × × 2 2

φ(xy) × × × × × × × × φ(xy) { a, b } × × {a, b, d} × × φ(xy) × × × × × ×

Lemma 10. All the monomials of T are positive and there is a one-to-one correspondence between the p-basis of P and the monomials of T . Proof. The proof consists of constructing a bijection, denoted by φ , from M to B. Obviously, by Lemma 9, M is the set of 0

t

0

s

0

q

monomials xy of T such x = α β γ and y = α β γ , where q, s, t are integers not necessarily zeros. Thus all monomials are positive, since all factors x are positive. Moreover, every monomial xy corresponds to a set X1 ∪ X2 and some cyclic flat F , such that x is the contribution of X1 in t (P |F ; 0, β, −1) and y is the contribution of X2 in t (P /F ; α, 0, γ ). Since X1 is the p-basis of P |F and X2 is the p-basis of P /F , so X1 ∪ X2 is a p-basis of P. We set φ(xy) = X = X1 ∪ X2 . Therefore φ is a function from M to B. Now, we need to show that φ is injective and surjective. (i) φ is surjective. Indeed, let B ∈ B. By Theorem 2, there is a cyclic flat F such that B ∩ F is a p-basis of P |F of zero internal activity and B\F is a p-basis of P /F of zero external activity. We aim to show that amongst all such cyclic flats, there is one with respect to which the contribution of B belongs to M (i.e., is not canceled out). For this purpose, set F to be the cyclic flat returned by Algorithm 1 when applied to B. By Theorem 2, B1 = B ∩ F and B2 = B \ F satisfy Condition (ii) of Lemma 9. Moreover, since F is the smallest cyclic flat containing B1 , so B1 is not augmentable. Hence, by Lemma 9, the contribution of B w.r.t. F belongs to M. (ii) φ is injective. Let xy and x′ y′ be two different monomials in M matched to the same p-basis B. Suppose that x and x′ , respectively, are monomials from t (P |F ; 0, β, −1) and t (P |F ′ ; 0, β, −1), where F and F ′ are two different cyclic flats. But this means that both F and F ′ are the smallest cyclic flats containing B1 . Hence F = F ′ , which is a contradiction.  Proof of Theorem 4. By Lemma 10, we have T =

 B

β

e(B)

α

i(B)

γ

r (P )−|B|

, and the proof follows.



Theorem 2 extends Theorem 1 given in [5] while Theorem 4 extends Theorem 3 given in [6]. Indeed, setting P = (M , M ) in both theorems, recovers both the results. Acknowledgments The authors are highly grateful to the anonymous referees for their valuable remarks and suggestions which improved the presentation of the paper.

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We dedicate this work to the memory of Michel Las Vergnas who inspired us many times and who helped us to improve this paper. Since the present article is a follow-up of [14], the authors of [14] wish also to point out an error in not acknowledging in [14] that the 4-variable linking polynomial Q (M , N ) of two matroids M, N defined in [14] is equivalent to the 3-variable Tutte polynomial of a matroid perspective t (M , N ) introduced by Las Vergnas in [5]. Quoting from p. 973 of [11]; ‘‘The classical Tutte polynomial is equivalent up to a simple transformation to the generating function of cardinality and rank of subsets of elements. The 3-variable Tutte polynomial studied in the present paper is in a similar way equivalent to the generating function of cardinality and ranks of subsets in both matroids. The assumption that the matroids are related by a strong map is necessary for certain properties to hold’’. To show this correspondence between t (M , N ) and Q (M , N ) one needs to extend the definition of the matroid perspective t (M , N ) via its rank generating function so that it is well defined for all matroid pairs (M , N ). With this extended definition it is easy to see that Q (M , N ) and t (M , N ) differ only by a simple multiplicative factor. Details of this transformation are given on page 394 of [14] for the case (M , N ) a matroid perspective. Using this correspondence it is straightforward to see that Theorem 3 and its corollaries from [14] are direct consequences of Theorem 8.1 of [11] which was first announced in [9]. The authors of [14] are very grateful to Michel Las Vergnas for pointing this out. References [1] D. Benard, A. Bouchet, A. Duchamp, On the Martin and Tutte polynomials, Technical Report, Département dInformatique, Université du Maine, Le Mans, France, 1997. [2] E.D. Bolker, H. Crapo, How to brace a one-story building, Environ. Plann. B Plann. Des. 4 (2) (1977) 125–152. [3] T. Brylawski, A combinatorial perspective on the radon convexity theorem, Geom. Ded. 5 (1976) 459–466. [4] J.A. Ellis-Monaghan, I. Moffatt, The las vergnas polynomial for embedded graphs, Eur. J. Comb. 50 (2015) 97–114. [5] G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, Discrete Math. 179 (1998) 111–119. [6] W. Kook, V. Reiner, D. Stanton, A convolution formula for the tutte polynomial, J. Combin. Theory Ser. B 76 (2) (1999) 297–300. [7] M. Las Vergnas, Extensions normale d’un matroide, polynôme de tutte d’un morphisme, C. R. Acad. Sci, Paris A 280 (1975) 1479–1482. [8] M. Las Vergnas, Acyclic and totally cyclic orientations of combinatorial geometries, Discrete Math. 20 (1977) 51–61. [9] M. Las Vergnas, On the tutte polynomial of a morphism of matroids, Ann. Disc. Math. 8 (1980) 7–20. [10] M. Las Vergnas, Eulerian circuits of 4-valent graphs imbedded in surfaces, in: L. Lovasz, V. Sos (Eds.), Algebraic Methods in Graph Theory, North-Holland, 1981, pp. 451–478. [11] M. Las Vergnas, The tutte polynomial of a morphism of matroids i. set-pointed matroids and matroid perspectives, Ann. Inst. Fourier, Grenoble 49 (3) (1999) 973–1015. [12] J.G. Oxley, Matroid Theory, Oxford University Press, New-York, 1992. [13] A. Recski, Maps of matroids with applications, Discrete Math. 303 (2005) 175–185. [14] D.J.A. Welsh, K.K. Kayibi, A linking polynomial of two matroids, Adv. Appl. Math. 32 (2004) 391–419. [15] T. Zaslavsky, Facing Up to Arrangemens: Face-Count Formulas for Partitions of Spaces by Hyperplanes, Mem. Amer. Math. Soc., Vol. 1, 1975, p. 154. Issue 1.