On Matroid Connectivity.

Certain classes of 2- and 3-connected matroids are studied in this thesis. In Chapter 2 we give a characterization of those 2-connected matroids M with the property that, for a given positive integer m, the deletion of every non-empty subset of M having at most m elements is disconnected. A bound on the maximum number of elements of such a matroid in terms of its rank is also given, along with a complete description of the matroids attaining this bound. These results extend results of Murty and Oxley for minimally 2-connected matroids. A characterization of the 3-connected matroids M that have the property that every 2-element deletion of M is disconnected is given in Chapter 3. It is shown that these matroids are exactly the duals of Sylvester matroids having at least four elements. In Chapter 4 we prove the following result: Let M be a 3-connected matroid other than a wheel of rank greater than three, and let C be a circuit of M. If the deletion of every pair of elements of C is disconnected, then every pair of elements of C is contained in a triad of M. For an integer t greater than one, an n-element matroid M is t-cocyclic if every deletion having at least n $-$ t + 1 elements is 2-connected, and every deletion having exactly n $-$ t elements is disconnected. A matroid is t-cyclic if its dual is t-cocyclic. In Chapter 5 we investigate the matroids that are both t-cocyclic and t-cyclic. It is shown that these matroids are exactly the uniform matroids U (t,2t) and the Steiner Systems S(t, t + 1, 2t + 2)

On matroid connectivity

If M is a loopless matroid in which M/vbX and M/vbY are connected and X∩Y is non-empty, then one easily shows that M/vb(X∪Y) is connected. Likewise, it is straightforward to show that if G and H are n-connected graphs having at least n common vertices, then G ∪ H is n-connected. The purpose of this note is to prove a matroid connectivity result that is a common generalization of these two observations. © 1995

Extremal Problems in Matroid Connectivity

Matroid k-connectivity is typically defined in terms of a connectivity function. We can also say that a matroid is 2-connected if and only if for each pair of elements, there is a circuit containing both elements. Equivalently, a matroid is 2-connected if and only if each pair of elements is in a certain 2-element minor that is 2-connected. Similar results for higher connectivity had not been known. We determine a characterization of 3-connectivity that is based on the containment of small subsets in 3-connected minors from a given list of 3-connected matroids. Bixby’s Lemma is a well-known inductive tool in matroid theory that says that each element in a 3-connected matroid can be deleted or contracted to obtain a matroid that is 3-connected up to minimal 2-separations. We consider the binary matroids for which there is no element whose deletion and contraction are both 3-connected up to minimal 2-separations. In particular, we give a decomposition for such matroids to establish that any matroid of this type can be built from sequential matroids and matroids with many fans using a few natural operations. Wagner defined biconnectivity to translate connectivity in a bicircular matroid to certain connectivity conditions in its underlying graph. We extend a characterization of biconnectivity to higher connectivity. Using these graphic connectivity conditions, we call upon unavoidable minor results for graphs to find unavoidable minors for large 4-connected bicircular matroids

Fork-decompositions of matroids

For the abstract of this paper, please see the PDF file

Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure

A notion of minor-based matroid connectivity

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M \backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we show that $U_{1,2}$ and $M(\mathcal{W}_2)$ are the only connected matroids $N$ such that, whenever a matroid has an $N$-minor using $\{e,f\}$ and an $N$-minor using $\{f,g\}$, it also has an $N$-minor using $\{e,g\}$. Finally, we show that $M$ is $U_{0,1} \oplus U_{1,1}$-connected if and only if every clonal class of $M$ is trivial.Comment: 13 page