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The Dilworth Number of Auto-Chordal Bipartite Graphs

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Abstract

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 \(k\).

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Acknowledgments

We are grateful to Van Bang Le for an interesting discussion leading to a simplified structure of the graphs \(D_k\).

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Authors and Affiliations

  1. LIMOS UMR CNRS 6158, Université Blaise Pascal, 63 173, Aubière, France

    Anne Berry

  2. Institut für Informatik, Universität Rostock, 18051, Rostock, Germany

    Andreas Brandstädt

  3. Institut für Mathematik, Universität Rostock, 18051, Rostock, Germany

    Konrad Engel

Authors
  1. Anne Berry
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  2. Andreas Brandstädt
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  3. Konrad Engel
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Correspondence to Konrad Engel.

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Berry, A., Brandstädt, A. & Engel, K. The Dilworth Number of Auto-Chordal Bipartite Graphs. Graphs and Combinatorics 31, 1463–1471 (2015). https://doi.org/10.1007/s00373-014-1471-8

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  • DOI: https://doi.org/10.1007/s00373-014-1471-8

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