How is a Chordal Graph like a Supersolvable Binary Matroid?

Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.Comment: 10 pages, 3 figures, to appear in Discrete Mathematic

A characterization of the base-matroids of a graphic matroid

Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases. A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$. The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ 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 that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti. Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$

A characterization of the base-matroids of a graphic matroid

Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases. A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$. The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ 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 that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti. Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$

How Is a Chordal Graph Like a Supersolvable Binary Matroid?

Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable i# G is chordal (rigid): this is an other way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M , a maximal chain of modular flats of M determines canonically a chordal graph

Linked Tree-Decompositions of Infinite Represented Matroids

It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids. Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width