Chordal matroids arising from generalized parallel connections II

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 (C₄),M(K₄)\. 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(K₄), M^*(K3,3), or M(Cₙ) for n≥ 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.