Chordal matroids arising from generalized parallel connections II

TL;DR

This paper characterizes $GF(2)$-chordal matroids as binary matroids avoiding certain minors and extends the concept of chordality from graphs to matroids via generalized parallel connections.

Contribution

It provides a complete characterization of $GF(2)$-chordal matroids and generalizes the perfect elimination ordering concept to $GF(q)$-chordal matroids.

Findings

$GF(2)$-chordal matroids are exactly binary matroids avoiding $M(K_4)$, $M^*(K_{3,3})$, and $M(C_n)$ for $n extgreater=4$.

Class of $GF(q)$-chordal matroids characterized by an analogous perfect elimination ordering.

Extension of chordal graph properties to matroids via generalized parallel connections.

Abstract

In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal matroids as those matroids that can be constructed from projective geometries over $GF(q)$ by a sequence of generalized parallel connections across projective geometries over $GF(q)$. Our main result showed that when $q=2$, such matroids have no induced minor in $\{M(C_4),M(K_4)\}$. In this paper, we show that the class of $GF(2)$-chordal matroids coincides with the class of binary matroids that have none of $M(K_4)$, $M^*(K_{3,3})$, or $M(C_n)$ for $n\geq 4$ as a flat. We also show that $GF(q)$-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices.