Supersolvable saturated matroids and chordal graphs

TL;DR

This paper explores supersolvable saturated matroids as analogues to chordal graphs, establishing key properties and structures like clique graphs and trees, and linking them to the concept of tree-width in matroids.

Contribution

It introduces the concept of supersolvable saturated matroids as chordal graph analogues and extends important graph-theoretic results to this matroid setting.

Findings

Supersolvable saturated matroids generalize chordal graphs.

Matroid analogues of clique trees are optimal for tree-width.

Key properties of chordal graphs extend to these matroids.

Abstract

A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroid context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of tree-width.