# Perfect Elimination Orderings for Symmetric Matrices

## TL;DR

This paper extends the concept of perfect elimination orderings from graphs to symmetric matrices, providing a unified framework for various structured matrices like chordal graphs, Robinsonian matrices, and ultrametrics.

## Contribution

It introduces a new class of structured matrices with perfect elimination orderings and characterizes them via forbidden substructures, generalizing known graph properties.

## Key findings

- Characterization of matrices with perfect elimination orderings
- Unified framework for chordal graphs, Robinsonian matrices, ultrametrics
- Structural forbidden substructure conditions

## Abstract

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.