A Dirac-type Characterization of k-chordal Graphs

TL;DR

This paper generalizes the characterization of chordal graphs to k-chordal graphs using notions of simplicial vertices and orderings, extending Dirac's classical results to a broader class of graphs.

Contribution

It introduces a new Dirac-type characterization of k-chordal graphs based on simplicial vertices and orderings, generalizing known results for chordal graphs.

Findings

Provides a characterization of k-chordal graphs using simplicial vertices.

Extends Dirac's classical characterization to k-chordal graphs.

Offers a new framework for understanding graph structure related to chordality.

Abstract

Characterization of k-chordal graphs based on the existence of a "simplicial path" was shown in [Chv{\'a}tal et al. Note: Dirac-type characterizations of graphs without long chordless cycles. Discrete Mathematics, 256, 445-448, 2002]. We give a characterization of k-chordal graphs which is a generalization of the known characterization of chordal graphs due to [G. A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25, 71-76, 1961] that use notions of a "simplicial vertex" and a "simplicial ordering".