Chordal matroids arising from generalized parallel connections

Abstract

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple GF(q)-representable matroids that can be built from projective geometries over GF(q) by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors. We characterize the class by its forbidden induced minors; the case when q=2 is distinctive.