The Dilworth Number of Auto-Chordal Bipartite Graphs.

The mirror (or bipartite complement) $${{\mathrm{mir}}}(B)$$ 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 $${{\mathrm{mir}}}(B)$$ if and only if $$xy \notin E$$ . 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 $$k$$ . [ABSTRACT FROM AUTHOR]