The Dilworth Number of Auto-Chordal-Bipartite Graphs

TL;DR

This paper introduces auto-chordal bipartite graphs, explores their properties, relationships to known graph classes, and characterizes those with specific Dilworth numbers, revealing they can have unbounded Dilworth number.

Contribution

It defines auto-chordal bipartite graphs and provides characterizations, relationships to other classes, and Dilworth number bounds, advancing understanding of this graph class.

Findings

ACB graphs have unbounded Dilworth number.

Characterization of ACB graphs with Dilworth number k.

Relationships established with interval and strongly chordal graphs.

Abstract

The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E) has the same color classes X and Y as B, and two vertices x in X and y in Y are adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal bipartite as well; we call these graphs auto-chordal bipartite graphs (ACB graphs for short). We describe the relationship to some known graph classes such as interval and strongly chordal graphs and we present several characterizations of ACB graphs. We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k.