A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutte's Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)−E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)−E(N)|?3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.     

The structure of the 3-separations of 3-connected matroids

Tutte defined a k-separation of a matroid M to be a partition (A,B) of the ground set of M such that |A|,|B|k and r(A)+r(B)−r(M)<k. If, for all m<n, the matroid M has no m-separations, then M is n-connected. Earlier, Whitney showed that (A,B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M.     

Removing circuits in 3-connected binary matroids

For a k-connected graph or matroid M, where k is a fixed positive integer, we say that a subset X of E(M) is k-removable provided M?X is k-connected. In this paper, we obtain a sharp condition on the size of a 3-connected binary matroid to have a 3-removable circuit.     

On the minor-minimal 3-connected matroids having a fixed minor

Let N be a minor of a 3-connected matroid M such that no proper 3-connected minor of M has N as a minor. This paper proves a bound on |E(M)−E(N)| that is sharp when N is connected.     

A Splitter Theorem Relative to a Fixed Basis

A standard matrix representation of a matroid M represents M relative to a fixed basis B, where contracting elements of B and deleting elements of E(M)–B correspond to removing rows and columns of the matrix, while retaining standard form. If M is 3-connected and N is a 3-connected minor of M, it is often desirable to perform such a removal while maintaining both 3-connectivity and the presence of an N-minor. We prove that, subject to a mild essential restriction, when M has no 4-element fans with a specific labelling relative to B, there are at least two distinct elements, s ₁ and s ₂, such that for each i ∈ {1, 2}, si(M/s _i ) is 3-connected and has an N-minor when s _i ∈ B, and co(M\s _i ) is 3-connected and has an N-minor when s _i ∈ E(M)–B. We also show that if M has precisely two such elements and P is the set of elements that, when removed in the appropriate way, retain the N-minor, then (P, E(M)–P) is a sequential 3-separation.     

Towards a splitter theorem for internally 4-connected binary matroids III

This paper proves a preliminary step towards a splitter theorem for internally 4-connected binary matroids. In particular, we show that, provided M   or M^?$M^{?}$ is not a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N   either has a proper internally 4-connected minor M^′$M^{\prime }$ with an N  -minor such that |E(M)−E(M^′)|?3$|E(M)-E(M^{\prime })|?3$ or has, up to duality, a triangle T and an element e of T   such that M\e$M\backslash e$ has an N-minor and has the property that one side of every 3-separation is a fan with at most four elements.     

On minor-minimally 3-connected binary matroids

A matroid M is called minor-minimally 3-connected if M is 3-connected and, for each e∈E(M), either M?e or M/e is not 3-connected. In this paper, we prove a chain theorem for the class of minor-minimally 3-connected binary matroids. As a consequence, we obtain a chain theorem for the class of minor-minimally 3-connected graphs.     

Elements belonging to triangles in 3-connected matroids

An element e of a 3-connected matroid M is said to be superfluous provided M/e is 3-connected. In this paper, we show that a 3-connected matroid M with exactly k superfluous elements has at least

     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Note on non-separating and removable cycles in highly connected graphs

We combine two well-known results by Mader and Thomassen, respectively. Namely, we prove that for any k-connected graph G (k≥4), there is an induced cycle C such that G−V(C) is (k−3)-connected and G−E(C) is (k−2)-connected. Both “(k−3)-connected” and “(k−2)-connected” are best possible in a sense.     

Long cycles in 4-connected planar graphs

Let G be a 4-connected planar graph on n vertices. Malkevitch conjectured that if G contains a cycle of length 4, then G contains a cycle of length k for every k∈{n,n−1,…,3}. This conjecture is true for every k∈{n,n−1,…,n−6} with k≥3. In this paper, we prove that G also has a cycle of length n−7 provided n≥10.     

5-Shredders in 5-connected graphs

For a graph G, a subset S of V(G) is called a shredder if G−S consists of three or more components. We show that if G is a 5-connected graph with |V(G)|≥135, then the number of shredders of cardinality 5 of G is less than or equal to (2|V(G)|−10)/3.     

Constructing a 3-tree for a 3-connected matroid

In an earlier paper with Whittle, we showed that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid M. The purpose of this paper is to give a polynomial-time algorithm for constructing such a tree for M.     

On the 3-Connected Matroids That are Minimal Having a Fixed Restriction

 Let N be a restriction of a 3-connected matroid M and let M ^′ be a 3-connected minor of M that is minimal having N as a restriction. This paper gives a best-possible upper bound on |E(M ^′)−E(N)|. Received: July 17, 1998 Revised: March 15, 1999     

Connectivity of iterated line graphs

Let k≥0 be an integer and L^k(G) be the kth iterated line graph of a graph G. Niepel and Knor proved that if G is a 4-connected graph, then κ(L²(G))≥4δ(G)−6. We show that the connectivity of G can be relaxed. In fact, we prove in this note that if G is an essentially 4-edge-connected and 3-connected graph, then κ(L²(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.     

Connected matroids with a small circumference

Lemos and Oxley proved that if M is a connected matroid with |E(M)|⩾3r(M), then M has a circuit C such that M⧹C is connected. In this paper, we shall improve this result proving that for a simple and connected matroid M, if r(M)⩾7 and |E(M)|⩾3r(M)−3, then M has a circuit C such that M⧹C is connected. To prove this result, we shall construct all the connected matroids having circumference at most five, with the exception of those which are 3-connected and have rank five.     

On the existence of nested orthogonal arrays

A nested orthogonal array is an OA(N,k,s,g) which contains an OA(M,k,r,g) as a subarray. Here r<s and M<N. Necessary conditions for the existence of such arrays are obtained in the form of upper bounds on k, given N, M, s, r and g. Examples are given to show that these bounds are quite powerful in proving nonexistence. The link with incomplete orthogonal arrays is also indicated.     

On some properties of base-matroids

Let M=(E,F) be a rank-n matroid on a set E and B one of its bases. A closed set θ⊆E is saturated with respect to B, or B-saturated, when |θ∩B|=r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I∩θ|?r(θ), for every closed B-saturated set θ, turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by M_B. In this paper we prove some properties of M_B, in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to M_B for every base B of M.Finally, we prove that the poset of the closed B-saturated sets ordered by inclusion is isomorphic to the Boolean lattice B_n.