article

The Dilworth Number of Auto-Chordal Bipartite Graphs

Authors:

Andreas Brandstädt,

Konrad EngelAuthors Info & Claims

Graphs and Combinatorics, Volume 31, Issue 5

Pages 1463 - 1471

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

Published: 01 September 2015 Publication History

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.

References

[1]

Benzaken, C., Hammer, P.L., de Werra, D.: Split graphs of Dilworth number 2. Discrete Mathematics 55, 123---127 (1985)

[2]

Berry, A., Sigayret, A.: Dismantlable lattices in the mirror. In: Cellier, P., Distel, F., Ganter, B. (eds.) Proceedings of ICFCA'13, LNAI 7880, pp. 44---59. Springer, Heidelberg (2013)

[3]

Berry, A., Brandstädt, A., Engel, K.: The Dilworth number of auto-chordal-bipartite graphs. arXiv:1309.5787vl (2013)

Digital Library

[4]

Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Math. Appl. SIAM, Philadelphia (1999)

Digital Library

[5]

Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Annals Discrete Mathematics 1, 145---162 (1977)

[6]

Dahlhaus, E.: Chordale Graphen im besonderen Hinblick auf parallele Algorithmen. Habilitation Thesis, Universität Bonn (unpublished) (1991)

[7]

Földes, S., Hammer, P.L.: Split graphs. Congr. Numer. 19, 311---315 (1977)

[8]

Földes, S., Hammer, P.L.: Split graphs having Dilworth number 2. Canadian J. Math. 29, 666---672 (1977)

[9]

Golumbic, M.C., Goss, C.F.: Perfect elimination and chordal bipartite graphs. JGT 2, 155---163 (1978)

[10]

Korpelainen, N., Lozin, V.V., Mayhill, C.: Split permutation graphs. Graphs and Combinatorics 30, 633---646 (2014)

Digital Library

[11]

Lubiw, A.: Doubly lexical orderings of matrices. SIAM J. Comput. 16, 854---879 (1987)

Digital Library

[12]

Nara, C.: Split graphs with Dilworth number three. Natural Science Report of the Ochanomizu University. 33(1/2), 37---44 (unpublished, available online) (1982)

[13]

Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16, 973---989 (1987)

Digital Library

[14]

Spinrad, J.P.: Efficient graph representations, Fields Institute Monographs, vol.19. AMS, Providence (2003)

[15]

Spinrad, J.P.: Doubly lexical ordering of dense 0---1 matrices. IPL 45, 229---235 (1993)

Digital Library

[16]

Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Alg. Discr. Meth. 3, 351---358 (1982)

The Dilworth Number of Auto-Chordal Bipartite Graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
    2. Graph theory
2. Theory of computation

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cover image Graphs and Combinatorics

Graphs and Combinatorics Volume 31, Issue 5

September 2015

691 pages

ISSN:0911-0119

EISSN:1435-5914

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Copyright © Copyright © 2015 Springer Japan.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 September 2015

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