The Dilworth Number of Auto-Chordal Bipartite Graphs

Abstract

The mirror (or bipartite complement) $${{\mathrm{mir}}}(B)$$mir(B) of a bipartite graph $$B=(X,Y,E)$$B=(X,Y,E) has the same color classes $$X$$X and $$Y$$Y as $$B$$B, and two vertices $$x \in X$$x?X and $$y \in Y$$y?Y are adjacent in $${{\mathrm{mir}}}(B)$$mir(B) if and only if $$xy \notin E$$xy?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 characterize ACB graphs, show that ACB graphs have unbounded bipartite Dilworth number, and we characterize ACB graphs with bipartite Dilworth number $$k$$k.