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 tight relation between series--parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by re17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum valu

T-uniqueness of some families of k-chordal matroids

We define k-chordal matroids as a generalization of chordal matroids, and develop tools for proving that some k-chordal matroids are T-unique, that is, determined up to isomorphism by their Tutte polynomials. We apply this theory to wheels, whirls, free spikes, binary spikes, and certain generalizations.Postprint (published version

Some inequalities for the Tutte polynomial

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.Comment: 17 page

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 $B$