Matroid Connectivity Research Articles

Matroid connectivity and singularities of configuration hypersurfaces

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

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 U1,2-connected. This paper proves that U1,2 is the only connected matroid N such that if M is N-connected with |E(M)|>|E(N)|, then M\\e or M/e is N-connected for all elements e. Moreover, we show that U1,2 and M(W2) are the only 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 U0,1⊕U1,1-connected if and only if every clonal class of M is trivial.

Matroid and Tutte-Connectivity in Infinite Graphs

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.

Intertwining Connectivity in Matroids

Let $M$ be a matroid and let $Q$, $R$, $S$, and $T$ be subsets of the ground set such that the smallest separation that separates $Q$ from $R$ has order $k$ and the smallest separation that separates $S$ from $T$ has order $\ell$. We prove that if $E(M)-(Q\cup R\cup S\cup T)$ is sufficiently large, then there is an element $e$ of $M$ such that, in one of $M\backslash e$ or $M/e$, both connectivities are preserved.

Minimally 3-connected binary matroids

A 3-connected matroid M is said to be minimally 3-connected if, for any element e of M , the matroid M ∖ e is not 3-connected. Dawes [R.W. Dawes, Minimally 3-connected graphs, J. Combin. Theory Ser. B 40 (1986) 159–168] showed that all minimally 3-connected graphs can be constructed from K 4 such that every graph in each intermediate step is also minimally 3-connected. Oxley [J.G. Oxley, On connectivity in matroids and graphs, Trans. Amer. Math. Soc. 265 (1981) 47–58] proved a similar result by giving a characterization of minimally 2-connected matroids. In this paper we generalize Dawes’ result to minimally 3-connected binary matroids. We give a constructive characterization of all minimally 3-connected binary matroids starting from W 3 , the 3-spoked wheel, and F 7 ∗ , the Fano dual.

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.

Some Small Circuit-Cocircuit Ramsey Numbers for Matroids

Ramsey numbers for matroids, which mimic properties of Ramsey numbers for graphs, have been denned as follows. Let k and l be positive integers. Then n(k, l) is the least positive integer n such that every connected matroid with n elements contains either a circuit with at least k elements or a cocircuit with at least l elements. We determine the largest known value of these numbers in the sense of maximizing both k and l. We also find extremal matroids with small circuits and cocircuits. Results on matroid connectivity, geometry, and extremal matroid theory are used here.

Whitney Connectivity of Matroids

A new definition of matroid connectivity is introduced and its properties are investigated in this paper. Vertex connectivity of graphs is expressed in an algebraic form and generalized to matroids. This generalized connectivity is called the Whitney connectivity of matroids. It is shown that the Whitney connectivity of the polygon matroid of a graph is the same as the vertex connectivity of the graph provided the graph is connected. Various properties of Whitney matroid connectivity and comparison with Tutte connectivity are also examined.

On Connectivity in Matroids and Graphs

In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer’s result that every minimally $2$-connected graph is $3$-colourable.

On matroid connectivity

Three types of matroid connectivity, including Tutte's, are defined and shown to generalize corresponding notions of graph connectivity. A theorem of Tutte on cyclically 3-connected graphs, is generalized to matroids.

On connectivity in matroids and graphs

In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer’s result that every minimally $2$-connected graph is $3$-colourable.

Theorems on matroid connectivity

Theorems on matroid connectivity

Connectivity in Matroids

An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only 3-connected graphs in which every edge is essential are the “wheel-graphs.” A wheel-graph of order n, where n is an integer ⩾3, is constructed from an n-gon called its “rim” by adding one new vertex, called the “hub,” and n new edges, or “spokes” joining the new vertex to the n vertices of the rim; see Figure 4A.A matroid can be regarded as a generalized graph. One way of developing the theory of matroids is therefore to generalize known theorems about graphs. In the present paper we do this with the theorem stated above.