# A separator-based method for generating weakly chordal graphs

## TL;DR

This paper introduces a new separator-based algorithm for efficiently generating weakly chordal graphs with specified numbers of vertices and edges, maintaining weak chordality throughout the process.

## Contribution

It presents a novel separator-based method for constructing weakly chordal graphs that ensures the property is preserved during edge insertions, improving over existing algorithms.

## Key findings

- The algorithm guarantees the graph remains weakly chordal after each insertion.
- Insertion query time complexity is O(n^3).
- Compared to existing methods, it offers a more efficient generation process.

## Abstract

We propose a scheme for generating a weakly chordal graph on n vertices with m edges. In this method, we first construct a tree and then generate an orthogonal layout (which is a weakly chordal graph on the n vertices) based on this tree. In the next and final step, we insert additional edges to give us a weakly chordal graph on m edges. Our algorithm ensures that the graph remains weakly chordal after each edge is inserted. The time complexity of an insertion query is O(n^3) time and an insertion takes constant time. On the other hand, a generation algorithm based on finding a 2-pair takes O(nm) time using the algorithm of Arikati and Rangan [1].