On the structure of triangle-free 3-connected matroids

Let N be an arbitrary class of matroids, closed under isomorphism. For k a positive integer, we say that M ∈ N is k-minor-irreducible if M has no minor N ∈ N such that 1 ≤ |E (M)| − |E (N)| ≤ k. Tutte’s Wheels and Whirls Theorem establish that, up to isomorphism, there are only two families of 1-minor-irreducible matroids in the class of 3-connected matroids. More recently, Lemos classified the 3-minor-irreducibles with at least 14 elements in the class of triangle-free 3-connected matroids. Here we prove a local characterization for the 2-minor-irreducible matroids with at least 11 elements in the class of triangle-free 3-connected matroids. This local characterization is used to establish two new families of 2-minor-irreducible matroids in this class.


Introduction
For arbitrary matroids M and N , N < M means that N is isomorphic to a proper minor of M . As usual, E (M ) denotes the ground set of M and M N means that M and N are isomorphic matroids. In this paper, F denotes the class of triangle-free 3-connected matroids.
Take M ∈ N , where N denotes an arbitrary class of matroids closed under isomorphism. We say that M is k-minor-reducible in N , for k a positive integer, if M has a minor N ∈ N such that 1 ≤ |E (M )| − |E (N )| ≤ k.
Since our focus is on triangle-free 3-connected matroids, we adopt the following convention: for k a positive integer, a matroid M is said to be k-minor-irreducible, without specifying family, if M is k-minor-irreducible in F. To contextualize the reader, we begin by stating the main results of Lemos [5,6] from the viewpoint of minor-irreducibility. Using certain reduction operations, Lemos [5] determines the 3-minor-irreducible matroids with at least 14 elements. In Section 2 we list the 2-minor-irreducible matroids mentioned in this introduction. Theorem 1.1. (compare with Theorem 1.7 - [5]) Let M be a 3-minorirreducible matroid with at least 14 elements. Then M is isomorphic to: i) an almost-double-wheel or an almost-double-whirl having rank at least 8; ii) the graphic matroid of a double-wheel with odd rank exceeding 7, or to a matroid obtained from a triadic Möbius matroid with even rank exceeding 8 after deleting it's tip; iii) a (m, n)-triangular matroid, for some non-negative integers m and n with m + n ≥ 2.
With this, our attention turns to the 2-minor-irreducible matroids that are 3-minor-reducible. Triads and squares, 4-set circuits, plays a fundamental role in the structure of 2-minor-irreducible matroids. The existence of a triad contained in a square is sufficiently restrictive, as shown by Theorem 1.2. A matroid M is said to be semi-binary provided T * Q, for every triad T * and square Q of M . Otherwise, M is said non-semi-binary. Theorem 1.2. (compare with Theorem 1.4 - [5]) Let M be a non-semi-binary 2-minor-irreducible matroid with at least 11 elements. Then: i) M is isomorphic to an almost-double-wheel or an almost-double-whirl; or ii) M is isomorphic to a non-binary ladder or to a relaxed non-binary ladder.
The main result of [6] establish that: Theorem 1.3. (compare with Theorem 1.2 - [6]) Let M be a 2-minorirreducible matroid with at least 11 elements. If there is an element e in exactly two triads such that co (M \e) is a triangle-free 3-connected matroid then: i) M is isomorphic to M (G), where G is a ladder or a Möbius ladder graph, if M is semi-binary; or ii) M is isomorphic a non-binary ladder or to a relaxed non-binary ladder, if M is non-semi-binary.
One of the main result of this paper is a local characterization for semibinary 2-minor-irreducible matroids. It establishes that each element belongs to one of the three configurations of triads and squares described below. Such structures, with the exception of the emerald, are named by Lemos in [5]. They are like 'building-blocks' for semi-binary 2-minor-irreducible matroids.
Sapphire: Let Q 1 and Q 2 be distinct squares of M such that |Q 1 ∩ Q 2 | = 1. If Q 1 ∩ Q 2 belongs to at least 2 triads T * and T * of M , S = Q 1 ∪ Q 2 is said to be a sapphire with nucleus Q 1 ∩ Q 2 . A sapphire S is called pure when there is another triad T * containing Q 1 ∩ Q 2 such that |T * ∩ T * | = |T * ∩ T * | = |T * ∩ T * | = 1 and S is closed in both M and M * .
Emerald: Let Q 1 and Q 2 be distinct squares of M such that |Q 1 ∩ Q 2 | = 2. If there are disjoint triads T * 1 and T * 2 of M such that T * 1 ∪ T * 2 = Q 1 ∪ Q 2 then E = Q 1 ∪ Q 2 is said to be an emerald. If the symmetric difference Q 1 Q 2 is also a square of M , we say that E is pure.
Diamond: Let Q 1 , Q 2 and Q 3 be squares of M such that |Q i ∩ Q j | = 1, for each 2-subset {i, j} of {1, 2, 3}, and Q 1 ∩Q 2 ∩Q 3 = ∅. If T * = (Q 1 ∩ Q 2 )∪ (Q 1 ∩ Q 3 ) ∪ (Q 2 ∩ Q 3 ) is a triad of M , D = Q 1 ∪ Q 2 ∪ Q 3 is said to be a diamond with nucleus T * . If T * does not intersects any other triad of M , we say that D is pure.
We are now ready to present one of the main result of this paper. Its proof is in Section 5. Theorem 1.4. Let M be a semi-binary 2-minor-irreducible matroid with at least 11 elements. For each e ∈ E (M ) there is a triad T * containing e such that T * is contained in a sapphire, or it is contained in an emerald or it is a nucleus of a pure diamond. Theorems 1.1 to 1.3 determine which are the 2-minor-irreducible matroids that avoid emeralds and pure diamonds, with at least 14 elements. Next result, established in Section 6, deals with 2-minor-irreducible matroids that avoid sapphires and emeralds. We denote by D the class of 2-minorirreducible matroids with at least 11 elements that avoids sapphires and emeralds. A matroid M ∈ D is called diamantic matroid. Theorem 1.5 establishes a bijection between this class of matroids and the class of totally triangular matroids. A 3-connected matroid M is called totally triangular if each of its elements belongs to at least 2 triangles, every pair of triangles intersects in at most 1 element and M has no triads.
Note that if M is totally triangular then M * is a triangle-free 3-connected matroid such that each element belongs to at least 2 triads and each pair of triads of M * intersects in at most 1 element. We denote by T the class of totally triangular matroids. Theorem 1.5. There is a bijection : D −→ T such that if M is a rank m diamantic matroid with n triads then (M ) is a totally triangular matroid with rank m − n and n triangles. Conversely, if M is a rank m totally triangular matroid with n triangles then −1 (M ) is a diamantic matroid with n triads and rank n + m.
In Section 7, we prove the last result of this paper. A sequence of equivalences on 3-connected matroids covered by pure emeralds.
Theorem 1.6. The following statements are equivalents for a 3-connected matroid M , with |E (M )| ≥ 9, in which each of its elements belongs to a pure emerald: i) every pair of elements is in a square; ii) for distinct elements x and y there is a pure emerald containing both; iii) M is binary; iv) M M (K 3,n ) where |E (M )| = 3n. This theorem has as a particular case one of the main results of a recent paper due to Oxley, Pfeil, Semple and Whittle [14], in which they establish the equivalence between (i) and (iv). As corollary of Theorem 1.6, we have: Theorem 1.7. Let M be a non-empty 2-minor-irreducible binary matroid such that each element belongs to an emerald. Then M M (K 3,n ) for some n ≥ 3.

Known families of 2-minor-irreducible matroids
In this section we list all known families of 2-minor-irreducible matroids.
The subset I = {i ∈ E : i is odd} is a circuit-hyperplane of one of these matroids, which we shall denote by M , called an almost-double-wheel. The matroid obtained from M relaxing the circuit-hyperplane I is called an almost-double-whirl. Moreover, we have that r (M ) = m + 1, I is a Hamiltonian circuit of M * and P = {i ∈ E : i is even} is an independent-hyperplane of M . The almost-double-wheel and almost-double-whirl are non-semi-binary 3-minor-irreducible matroids defined and constructed by Lemos in Section 5 of [5]. Follows an auxiliary graph to illustrate these squares and triads: Figure 1: The 3-set of edges incident with vertices of degree 3 illustrate the triads, the edges set of a 4-cycle in this graph represents the squares mentioned in (iii) and the 4-set of dashed lines is a square containing the triad {1, 2, 3}.

Double-wheel and Triadic Möbius matroid:
Let M (W n ) be a n-wheel with rank n ≥ 6. There is just one 3-connected binary matroid N with ground set E (N ) = E (M (W n )) ∪ {e}, for a new element e, such that N is triangle-free and N/e = M (W n ). When n is even, N is graphic and D n = N \e is called double-wheel with rank n + 1. When n is odd, the matroid N is called triadic Möbius matroid and the element e called tip of N . Mayhew, Royle and Whittle [8] denoted N by Υ n+1 . Its rank is n + 1. Double-wheel and triadic Möbius matroid with its tip deleted are both semi-binary 3-minor-irreducible matroids.

(m, n)-triangular matroid:
A 3-connected matroid M is said to be (m,n)-triangular, for non-negative integers m and n such that m+n ≥ 2, when M is obtained from a matroid N whose ground set is partitioned into m+n triangles, say T 1 , . . . , T m , T 1 , . . . T n , and whose simplification is 3-connected by: (i) adding an element e in series with each element e of N ; and (ii) for each i ∈ {1, . . . , m}, adding an element e i such that, for every e ∈ T i , {e i , e, e } is a triad of M ; and (iii) for each i ∈ {1, . . . , n}, adding elements e i , f i , g i such that Rubies: Suppose that there is a triad T * and pairwise disjoint triads where the indices are taken modulus 3. Then R = T * 0 ∪ T * 1 ∪ T * 2 is said to be a ruby of M with nucleus T * . We say that R is pure provided is closed in both M and M * .
The ground set of a (m, n)-triangular matroid M is partitioned into m pure sapphires, S i for i ∈ {1, ..., m}, and n pure rubies, R j for j ∈ {1, ..., n}.

Ladder and Möbius ladder:
For n ≥ 4, the ladder L n with 2n vertices is the graph illustrated in Figure  4. We denote by M (L n ) its cycle matroid. If in L n we delete the edges {T * 1 , T * n } and {T 1 * , T n * } and we add new edges {T 1 * , T * n } and {T * 1 , T n * }, then we get the Möbius ladder graph L n with 2n vertices. We denote by M (L n ) its cycle matroid.

Non-binary ladder and relaxed non-binary ladder:
For n ≥ 4, let G n be the auxiliary graph displayed in Figure 5. Set D = {a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n }. Then the relaxed non-binary ladder R n of rank 2n is a matroid over E (G n ) such that C (R n ) = C ∪ D, where i) C ∈ C if and only if C is a circuit of the cycle matroid associated with G n and C = D ∪ {c 0 , c n }; and ii) C ∈ D if and only if C = E (T ), where T is a tree of G n such that: each leaf vertex of T is incident in G n with c 0 or c n , and every vertex incident with c 0 or c n in G n is a vertex of T . The 2n-set D is a basis of R n . There is a matroid P n over E (G n ) such that and R n is obtained from P n by relaxing the circuit-hyperplane D. We say that P n is the non-binary ladder of rank 2n. The non-binary-ladder and relaxed non-binary-ladder are both non-semi-binary 2-minor-irreducible matroids constructed by Lemos in Section 2 of [6].

Diamantic matroids:
A 3-connected matroid M is called a diamantic matroid if each of its elements belongs to a nucleus of a pure diamond and M has no emeralds. There is a bijection from the class of diamantic matroids to the class of totally triangular matroids such that for each rank m diamantic matroid M with n triads, (M ) is a totally triangular matroid with rank m − n and n triangles. Since P 7 , displayed in Figure 8, is the totally triangular matroid with fewest triangles, we have −1 (P 7 ) the smallest diamantic matroid. Follows a graph representation of a diamantic matroid and its totally triangular matroid associated. Figure 6: The cycle matroid of G is a diamantic matroid such that its totally triangular matroid associated is the cycle matroid of (G). Degree 3 vertices of G are pointed out.
2.7. The cycle matroid of bipartite graph M (K 3,n ), for n ≥ 3: In this context of minor-irreducibility, this is the family of binary 2-minorirreducible matroids such that each element belongs to an emerald. Note that M (K 3,3 ) is a 3-minor-irreducible matroid.

Preliminary results
We use the terminologies and notation set in [12]. If M denotes a matroid, its connectivity function is where r is the rank function of M and r * the rank function of M * . A subset

Knowns results on 3-connected matroids.
We start with some key results on 3-connected matroids. From Lemos [4], we use the following result: From Oxley [10], we use the following result: Suppose that e and f are distinct elements of a n-connected Then M has a cocircuit with length n containing e and f .

Reduction operations on triangle-free 3-connected matroids.
Consider the following reduction operations on F, operations that when applied to the elements of a matroid M ∈ F produce a minor N < M and N ∈ F. The first two reduction operations are: A triangle-free 3-connected matroid M is called 12-irreducible if it is both 1-irreducible and 2-irreducible. Therefore, M is 12-irreducible if and only if M is 1-minor-irreducible in F. Classifying 1-minor-irreducible matroids in F is an unviable task. For this reason, we shall consider another reduction operation: • Third reduction: A triangle-free 3-connected matroid M is called 3-reducible when there are squares Q 1 and Q 2 intersecting in a single element, say f , belonging to a unique triad such that A triangle-free 3-connected matroid M is called 123-irreducible if it is iirreducible for every i ∈ {1, 2, 3}. Despite the similarity between i-irreducible and i-minor-irreducible notations, for 1 ≤ i ≤ 3, we decided to keep the notation used by Lemos [5]. We hope it does not cause confusion for the reader. The following lemmas are both in Section 2 of Lemos [5].
Lemma 3.7. Suppose that M is a semi-binary 2-irreducible matroid. Then each coline of M has at most 3 elements.

Fullclosure operator and sequential separation.
The terminologies for fullclosure operator and sequential separations were introduced by Oxley, Semple and Whittlel [13]. Let M be a matroid. We define the fullclosure operator as the function f cl M : where cl and cl * denotes, respectively, the closure and coclosure operator of M . Note that f cl M (X) = f cl M * (X). We denote by f cl (X) when it does not cause confusion.
One way of obtaining the fullclosure of a subset X ⊆ E (M ) is to take alternately closure and coclosure and so on until neither the closure nor the coclosure operator adds new elements. The elements of f cl (X) − X can be ordered The following result hold for k-separating sets with k ≥ 1, but our only interest is in the case k = 2 and 3: iii) The elements of f cl (X) − X can be ordered   [5]) Suppose that M is a triangle-free 3-connected matroid and e ∈ E (M ). If {X, Y } is a non-trivial 2-separation for M \e then {X, Y } is non-sequential or e belongs to a coline with at least 4 elements. Moreover, when M is also semi-binary and 2-irreducible, {X, Y } is non-sequential.

Forced Sets.
This subsection contains some results set out in Section 4 of [5]. They are used to establish the auxiliary results in the next section. Let M be a matroid with ground set Forced sets are not separated by 2-separations on matroids resulting from contraction or deletion of elements outside F .
Lemma 3.14. Suppose that M is a triangle-free 3-connected matroid with at least 5 elements and F is a forced set of M .

Auxiliary lemmas on intersection of squares
The importance of squares and triads for triangle-free 3-connected matroids has already been mentioned. In this section, we have established four auxiliary lemmas dealing with certain configurations of squares and triads. These results are used in the next section, where we prove Theorem 1.4. Each subsection is dedicated to one of these configurations.

4.1.
A pair of squares having just one element in common.
otherwise there is a triad of M/e containing {f, x}. Using Bixby's Theorem 3.3, we have that there is a non-trivial (exact) 2-separation for M \f , say {X, Y }. Suppose that {X, Y } is sequential. Lemma 3.8 implies that we can put an order on X or Y , say Y = {y 1 , . . . , y n−2 , y n−1 , y n }, with n ≥ 3, such that {y n−2 , y n−1 , y n } and {y n−1 , y n } are both 2-sparating set for M \f . Then {y n−1 , y n } is in a series class of M \f , {y n−1 , y n } = {e, g}, and so {y n−2 , e, g, } is a triad of M . Hence y n−2 ∈ Q 1 ∩ Q 2 , because of orthogonality; a contradiction. Therefore {X, Y } is non-sequential.
Denote by T * x and T * y the triads of M that contains {e, x} and {e, y}, respectively, where {x, y} = Q 1 − T * . We can suppose that |X ∩ T * x | ≥ 2. If y ∈ X then f belongs to cl (X), contradicting 3-connectivity of M . So T * y ∩ Y ≥ 2, we have that g ∈ X and |T * ∩ Y | ≥ 2. Therefore

A pair of squares having two elements in common.
First, note that if T * is a triad of a semi-binary matroid M that intersects Q 1 ∪ Q 2 , where Q 1 and Q 2 are squares of M such that |Q 1 ∩ Q 2 | = 2 , then: Proof. Let T * and T * be distinct triads contained in F = Q 1 ∪ Q 2 . We have r (F ) ≤ 4, because of squares Q 1 and Q 2 , and r * (F ) ≤ 4, because of triads T * and T * . Hence λ (F ) ≤ 2 and |E (M ) − F | ≥ 3. The 3-connectivity of M implies that λ (F ) = 2 and so F is a 3-separating set with r (F ) = r * (F ) = 4. If |T * ∩ T * | = 2 then L * = T * ∪ T * is contained in a coline and so M has a square containing a triad; a contradiction.
Finally, we can assume that T * ∩ T * = ∅ and so F is an emerald.
which is a contradiction. Therefore |Y − T * | = 1 and |Y | = 2. There is a cocircuit C * of M such that Y = C * − T * . Since F is a 3-separating forced set of M and |C * − F | = 1, Lemma 3.16 implies that M is i-reducible for some i ∈ {1, 2, 3}; a contradiction.
Proof. Here, M denotes a semi-binary 123-irreducible matroid with at least 11 elements. Assume that Q 1 and Q 2 are squares of M such that |Q 1 ∩ Q 2 | = 2 and Q 1 ∪Q 2 is not an emerald. Denote by F = Q 1 ∪Q 2 . Using an argument similar to that found in Lemos [5], Section 8, we have: Sub-lemma 4.4.1. F contains at most one triad of M and there are disjunct triads T * i , for each i ∈ {1, 2, 3}, such that Sub-lemma 4.4.2. For each i ∈ {1, 2, 3} there are squares Q i containing e i such that: Suppose that (ii) does not occur. Assume, by contradiction, that Proof. Suppose, for i = 1, that T * is another triad, different from T * 1 , such that Q 1 ∩ Q 1 ⊆ T * . Since |T * ∩ T * 1 | ≤ 1, because of Lemma 3.7, T * is contained in F and |Q 1 ∩ F | > 1; a contradiction.   As M is 2-irreducible, there is a square Q 1 containing e i . If |Q i ∩ Q i | = 2, Q 1 ∪Q 1 = T * ∪T * is an emerald, because of Lemma 4.4. Suppose |Q i ∩ Q i | = 1 and take {g i } = Q i ∩ Q i . By Lemma 4.1, M \g i /e i is 3-connected and then there is a square Q i containing e i avoiding g i . We can suppose The same argument used in Lemma 6.3 of [5] shows that M \T * is 3-connected.

Proof of Theorem 1.4: local characterization for 2-minorirreducible matroids
This section contains a sequence of results that constitute the proof of Theorem 1.4. Here, M denotes a semi-binary 123-irreducible matroid with at least 11 elements. We denote by S the union of sapphires, pure diamonds and emeralds of M . Our goal is to show that S = E (M ). Proof. Suppose M/e is 3-connected. Since Q S, Lemma 5.1 implies that there are squares Q 1 and Q 2 such that T * i ⊆ Q ∪ Q i and Q ∩ Q i = {f i }, for i ∈ {1, 2}. Therefore T * i is the unique triad that contains f i and co (M \f i ) = M \f i /g is triangle-free, so is not 3-connected because M is 3irreducible. As co (M/e\f 1 ) = M/ {e, g} \f 1 is 3-connected, we have e is in a series class of M \f 1 /g; a contradiction.
Proof. As M/e is not 3-connected, Lemma 5.1 implies that there is a triad T * 3 that contains e. If g / ∈ T * 3 then its contains f i for some i ∈ {1, 2}. Suppose, without lost of generality, that f 1 ∈ T * 3 . Since {f 1 } = T * 1 ∩ T * 3 ∩ Q and Q S, we have that Q is the unique square containing f 1 and then plays the role of e in the previous sub-lemma. Hence M/f 2 = si (M/f 2 ) is not 3-connected and dual form of Tutte's Triangle Lemma 3.4 implies that M/g 2 and M/g are both 3-connected. Therefore M has a square Q that contains {f 2 , g 2 } and T * 2 is the unique triad that contains f 2 . As M is 3-irreducible, there is a square Q that contains g 2 and avoids f 2 . By orthogonality with T * 2 , we have g ∈ Q; a contradiction. Therefore T * 3 ∩ Q = {e, g}.
Denote by f 3 the element in T * 3 − Q. We have M/f 3 and M/g are both 3-connected, because of the dual form of Tutte's Triangle Lemma. There is a square Q of M such that f 3 ∈ Q and Lemma 4.4 implies that {e} = Q∩Q . Therefore T * 3 is the unique triad containing e and co (M \e) = M \e/f 3 is 3-connected. There is a square Q containing f 3 and avoiding e. By orthogonality with T * 3 we have g ∈ Q ; a contradiction. Lemma 5.3. If Q is a square of M such that Q ∩ T * 1 ∩ T * 2 = ∅ for every pair of distinct triads T * 1 and T * 2 of M then Q ⊆ S. As consequence, every square of M is contained in S.
Proof. Because of Theorem 3.1, there are triads T * i = {e i , f i , g i }, for i ∈ {1, 2}, such that Q = {f 1 , g 1 , f 2 , g 2 } and {e i } = T * i − Q. By hypothesis, they are the only triads intersecting Q. We can assume M/f i 3-connected.
Because of Bixby's Theorem 3.3, we have si In this case, Q ∪ Q is an emerald, because of Lemma 4.4, and Q ⊆ S. Suppose M/f i /e j is 3-connected, then M/e j is also 3-connected and, by 2-irreducibility of M , there is a square Q containing e j . If e 1 = e 2 , Q ∪ Q is an emerald and hence Q ⊆ S. If e 1 = e 2 , Lemma 5.1 implies that there are squares Q 1 and Q 2 such that T * i ⊆ Q ∪ Q i , for i ∈ {1, 2}. We can assume that |Q ∩ Q i | = 1, otherwise Q ∪ Q i is an emerald. Because of Lemma 4.1, co (M \ (Q ∩ Q i )) = M \ (Q ∩ Q i ) /e i is 3-connected. Since M is 3-irreducible, there are squares Q 1 and Q 2 such that Q i contains e i and avoid the element in Q ∩ Q i . Thus Lemmas 5.2 to 5.5 establish that each element of M belongs to a triad contained in union of sapphires, emeralds and pure diamonds. Next lemma improves this result.
Lemma 5.6. If T * is a triad of M then we have 3 possibilities: i) T * is contained in an emerald; or ii) T * is a nucleus of a pure diamond; or iii) there is a sapphire containing T * such that its nucleus is an element of T * .
Proof. Because of Lemma 4.3, every triad that intersects an emerald is contained in it. Suppose that T * intersects a pure diamond, but is neither a nucleus of a pure diamond nor intersects an emerald. Then Lemma 4.5 implies that T * contains a nucleus of a sapphire.
Suppose that T * is a triad of M such that T * is not a nucleus of pure diamond, do not contains a nucleus of a sapphire and do not intersects an emerald. Then each element of T * is contained in a sapphire S. So, there are distinct squares Q 1 and Q 2 such that |Q 1 ∩ Q 2 | = 1 and T * is the unique triad that contains Then there is a triad T * and a 2-subset {e, f } of T * such that N M \e/f . Since N is triangle-free, T * is neither a nucleus of a pure diamond nor is contained in an emerald. Therefore T * contains a nucleus of a sapphire and so N has a triangle; a contradiction.

Proof of Theorem 1.5: Diamantic and totally triangular matroids
In the first part of this section we give a simple way to check that P 7 (Figure 8) is the totally triangular matroid with fewest triangles. Then, we define an extension operation for 3-connected matroids, called triangulation around a triad, which is a process of 'placing a triangle around a triad'. In the end, we apply the triangulation around a triad on diamantic matroids to get an associated totally triangular matroid for each of them.

Totally triangular matroid with fewest triangles.
Tutte's Wheels and Whirls Theorem [17] implies that 3-connected binary matroids with at least 4 elements have M (W 3 )-minor, since it avoids U 2,4minor. There are only two 3-connected binary matroids with 7 elements: the Fano matroid F 7 is the binary totally triangular matroid with fewest triangles and F * 7 is the smallest binary triangle-free 3-connected matroid. Let M be a totally triangular matroid that is representable over some field and suppose that M is non-binary with at least 4 elements. Suppose |E (M )| ≤ 7. Because of Theorem 2.1 of [11], we can assume that M has no U 2,5 -minor. The main result of [15] implies that M is ternary. Theorem 2 of [1] implies that M P 7 or M has a minor isomorphic to a 3-wheel. For convenience, in this subsection we call a 3-connected matroid by strictly triangular matroid if each of its elements belongs to at least 2 triangles and every pair of triangles intersects in at most one element. So, a totally triangular matroid is a triad-free strictly triangular matroid. Lemma 6.1. Up to isomorphism: i) the 3-wheel M (W 3 ) is the only strictly triangular matroid with 4 triangles; and ii) P 7 is the only strictly triangular matroid with 5 triangles. As consequence, P 7 is the totally triangular matroid with fewest triangles.

Triangulation around a triad.
The bijection referred in Theorem 1.5 is based on a construction via generalized parallel connection. For definitions, notations and properties on generalized parallel connection see Oxley's book [12], sections 11.4 and 11.5. Let M be a 3-connected matroid, T * a triad and Y a triangle of M . We say that Y is around T * in M if M | (T * ∪ Y ) is isomorphic to a cycle matroid of a 3-wheel.
Denote by K 5 \e the complete graph on five vertices with one edge deleted. We call M * (K 5 \e) by prism matroid. The prism matroid has two disjoint triangles, say T and T . If M is a 3-connected matroid and T * is The following results derive from the properties of the generalized parallel connection and usual arguments in matroids. We have omitted their respective proofs. 6.3. One-to-one correspondence between diamantic and totally triangular matroids.
Let M be a diamantic matroid with n triads. Fix an order for the family of triads of M , say {T * 1 , T * 2 , . . . , T * n }. For each k ∈ {1, 2, . . . , n}, we denote by W k 3 a copy of a 3-wheel with ground set E W k , for each k ∈ {2, . . . , n}. Consider the following sets Lemma 6.5. For each k ∈ {1, 2, . . . , n}, M k is a 3-connected matroid such that: is a 3-connected matroid with rank r (M ) − k; iv) each element of Y 1 ∪ · · · ∪ Y k belongs to at least two triangles of M k \ (T * 1 ∪ · · · ∪ T * k ); v) the triads of M k are the same triads of M .
Proof. Items (i ), (ii ) and (iii ) are consequence of Lemma 6.3. Item (iv ): suppose valid for k − 1. Take y ∈ Y 1 ∪ · · · ∪ Y k . If y ∈ Y j for j < k then y belongs to at least two triangles of M k−1 , and so of M k . Otherwise y ∈ Y k and we can apply the same arguments as Lemma 6.3 (iii). Item (v ) is consequence of Lemma 6.3 (v), since each element of M belongs to a unique triad.
It is straightforward to see that M n does not depend on the order of the triads.
Denote by D n the family of diamantic matroids with n triads and T n the family of totally triangular matroids with n triangles, for n ≥ 5 (Lemma 6.1). Previous lemma implies that : D n −→ T n such that is an injective function, up to isomorphism. The construction of reveals how to obtain its inverse. From a totally triangular matroid N , the process for obtaining the associated diamantic matroid is simpler as it involves general parallel connection directly.

Proof of Theorem 1.6: (ii) implies (iv)
We need the concept of weak map. Let M and N be two matroids and let ω : E (M ) −→ E (N ) be a bijection. We say that ω is a weak map from M to N if for each independent set I in N , we have ω −1 (I) is independent in M . Equivalently, for every circuit C of M , we have ω (C) contains a circuit of N . The following theorem is due to Lucas [7]. Proof. We can partition E (M ) in a disjunct union of triads E (M ) = T * 1 ∪ T * 2 ∪ · · · ∪ T * n for some n ≥ 3 such that T * i ∪ T * j is a pure emerald for 1 ≤ i < j ≤ n. Furthermore, we can choose the labels of T * i = {e i , f i , g i } so that {x i , y i , x j , y j } is a square of M for each 2-subset {x, y} ⊆ {e, f, g} and for 1 ≤ i < j ≤ n. By orthogonality, for distinct triads T * i , T * j and T * k , if Q is a square contained in T * i ∪ T * j and Q is a square contained in T * j ∪ T * k such that |Q ∩ Q | = 1, then the symmetric difference Q Q is a circuit of M . Lemma 4.2 implies that F = T * 1 ∪ T * 2 is a forced set with r (F ) = r * (F ) = 4. For each 2 < i ≤ n, T * i ∩ cl T * 1 ∪ · · · ∪ T * i−1 = ∅ and then r T * 1 ∪ · · · ∪ T * i−1 ∪ T * i = r T * 1 ∪ · · · ∪ T * i−1 + 1. Consequently r (M ) = n + 2.
Let N be a matroid with ground set E (N ) = E (M ) such that N M (K 3,n ) having {e i , f i , g i } as triads and having {x i , y i , x j , y j } as squares for each 2-subset {x, y} ⊆ {e, f, g} and for 1 ≤ i < j ≤ n. The identity map from E (N ) to E (M ) is a weak map from M (K 3,n ) to M . As r (M (K 3,n )) = n + 2 and M is 3-connected, Theorem 7.1 implies that M M (K 3,n ).

Proof of Theorem 1.6: (iii) implies (iv)
It is known that if M is a 3-connected binary matroid with 6 ≤ |E (M )| ≤ 8 then M is isomorphic to M (W 3 ) , F 7 , F * 7 , AG (3, 2) , S 8 or M (W 4 ). All of these matroids are well known. Let M be a 3-connected binary matroid with |E (M )| = 9. If M has no 4-wheel minor then M is isomorphic to Z 4 or Z * 4 . If M has a 4-wheel minor and is non-regular then M is isomorphic to P 9 or P * 9 . If M is a regular matroid having a 4-wheel minor then M is isomorphic to M (K 3,3 ) , M * (K 3,3 ) , M (K 5 \e) or M * (K 5 \e). Matrix representations of Z 4 and P 9 can be found in Kingan and Lemos [3]. where T * is a triad of M (K 3,n−1 ), T is the triangle around T * in T * (M (K 3,n−1 )) and W 3 is a 3-wheel over E (W 3 ) = T ∪ T * with rim T and E (W 3 ) ∩ E ( T * (M (K 3,n−1 ))) = T .
If there are elements of M \T * 1 that do not belongs to any emerald then these elements are in T where x ∈ F , t ∈ T and X is a parallel class in M 1 . As consequence, si (M 1 ) is 3-connected. These properties of M stem from the results on 3-separations due to Seymour [16]. For more direct results, see Proposition 9.3.4 [12] and (4.3) [16]. Every circuit C of M 2 such that |C ∩ T | = 1 has odd cardinality and then si (M 1 ) = M 1 , otherwise M 1 has a parallel class {x, t} where x ∈ F and t ∈ T . In this case M = P T (M 1 , M 2 ) \T has a circuit with odd cardinality, and this is a contradiction.