Chordal matroids arising from generalized parallel connections II
James Dylan Douthitt, James Oxley
TL;DR
This work extends Dirac-style chordality from graphs to matroids by using generalized parallel connections across projective geometries over $GF(q)$. It establishes a precise forbidden-minor/restriction framework for $GF(2)$-chordal matroids, showing they are exactly the binary matroids with no induced minor from $\\{M(C_4),M(K_4)\\}$ and with no induced restriction from $\{M(C_n):n\ge4\\}\\cup\\\{M(K_4),M^*(K_{3,3})\\}$; it further generalizes to $GF(q)$ with corresponding forbidden families of uniform matroids, depending on $q$. A key conceptual link is drawn to Rose's perfect elimination ordering by proving that $GF(q)$-chordality is equivalent to the existence of a perfect elimination ordering of cocircuits, and that projective geometries admit such an ordering. The results together provide a structural, constructive framework for chordal matroids across different fields, paralleling the graph-theoretic characterization of chordal graphs. These findings yield a clear, algebraic analogue of Dirac’s and Rose’s classical results in the matroid setting, with practical implications for recognizing and decomposing GF($q$)-chordal matroids via cocircuit orderings and modular-connection decompositions.
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.