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Restricted Bipartite Graphs: Comparison and Hardness Results

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Algorithmic Aspects in Information and Management (AAIM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8546))

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Abstract

Convex bipartite graphs are a subclass of circular convex bipartite graphs and chordal bipartite graphs. Chordal bipartite graphs are a subclass of perfect elimination bipartite graphs and tree convex bipartite graphs. No other inclusion among them is known. In this paper, we make a thorough comparison on them by showing the nonemptyness of each region in their Venn diagram. Thus no further inclusion among them is possible, and the known complexity results on them are incomparable. We also show the \(\mathcal{NP}\)-completeness of treewidth and feedback vertex set for perfect elimination bipartite graphs.

Partially supported by National 973 Program of China (Grant No. 2010CB328103) and Natural Science Foundation of China (Grant Nos. 61370052 and 61370156).

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References

  1. Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8, 277–284 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brandstad, A., Le, V.B., Spinrad, J.P.: Graph Classes - A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Google Scholar 

  3. Golumbic, M.C., Goss, C.F.: Perfect elimination and chordal bipartite graphs. J. Graph Theory 2, 155–163 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Grover, F.: Maximum matching in a convex bipartite graph. Nav. Res. Logist. Q. 14, 313–316 (1967)

    Article  Google Scholar 

  5. Jiang, W., Liu, T., Ren, T., Xu, K.: Two hardness results on feedback vertex sets. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 233–243. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. Jiang, W., Liu, T., Wang, C., Xu, K.: Feedback vertex sets on restricted bipartite graphs. Theor. Comput. Sci. 507, 41–51 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jiang, W., Liu, T., Xu, K.: Tractable feedback vertex sets in restricted bipartite graphs. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 424–434. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  8. Karp, R.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  9. Kloks, T.: Treewidth: Computations and Approximations. Springer (1994)

    Google Scholar 

  10. Kloks, T., Kratsch, D.: Treewidth of chordal bipartite graphs. J. Algorithms 19, 266–281 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kloks, T., Liu, C.H., Pon, S.H.: Feedback vertex set on chordal bipartite graphs. arXiv:1104.3915 (2011)

    Google Scholar 

  12. Kloks, T., Wang, Y.L.: Advances in Graph Algorithms (2013) (manuscript)

    Google Scholar 

  13. Liang, Y.D., Blum, N.: Circular convex bipartite graphs: Maximum matching and Hamiltonian circuits. Inf. Process. Lett. 56, 215–219 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, T., Lu, Z., Xu, K.: Tractable connected domination for restricted bipartite graphs. J. Comb. Optim. (2014), , doi 10.1007/s10878-014-9729-x

    Google Scholar 

  15. Lu, M., Liu, T., Xu, K.: Independent domination: Reductions from circular- and triad-convex bipartite graphs to convex bipartite graphs. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 142–152. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  16. Lu, M., Liu, T., Tong, W., Lin, G., Xu, K.: Set cover, set packing and hitting set for tree convex and tree-like set systems. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds.) TAMC 2014. LNCS, vol. 8402, pp. 248–258. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  17. Lu, Z., Liu, T., Xu, K.: Tractable connected domination for restricted bipartite graphs (extended abstract). In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 721–728. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  18. Lu, Z., Lu, M., Liu, T., Xu, K.: Circular convex bipartite graphs: Feedback vertex set. In: Widmayer, P., Xu, Y., Zhu, B. (eds.) COCOA 2013. LNCS, vol. 8287, pp. 272–283. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  19. Song, Y., Liu, T., Xu, K.: Independent domination on tree convex bipartite graphs. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds.) AAIM 2012 and FAW 2012. LNCS, vol. 7285, pp. 129–138. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  20. Tucker, A.: A structure theorem for the consecutive 1’s property. Journal of Combinatorial Theory 12(B), 153–162 (1972)

    Google Scholar 

  21. Wang, C., Chen, H., Lei, Z., Tang, Z., Liu, T., Xu, K.: Tree convex bipartite graphs: \(\mathcal{NP}\)-complete domination, hamiltonicity and treewidth. In: Proc. of FAW (2014)

    Google Scholar 

  22. Wang, C., Liu, T., Jiang, W., Xu, K.: Feedback vertex sets on tree convex bipartite graphs. In: Lin, G. (ed.) COCOA 2012. LNCS, vol. 7402, pp. 95–102. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  23. Yannakakis, M.: Node-deletion problem on bipartite graphs. SIAM J. Comput. 10, 310–327 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer Science, Peking University, Beijing, 100871, China

    Tian Liu

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  1. Tian Liu
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Editor information

Editors and Affiliations

  1. School of Computing Science, Simon Fraser University, V5A 1S6, Burnaby, BC, Canada

    Qianping Gu  & Pavol Hell  & 

  2. Department of Computer Science, University of Regina, S4S 0A2, Regina, SK, Canada

    Boting Yang

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Liu, T. (2014). Restricted Bipartite Graphs: Comparison and Hardness Results. In: Gu, Q., Hell, P., Yang, B. (eds) Algorithmic Aspects in Information and Management. AAIM 2014. Lecture Notes in Computer Science, vol 8546. Springer, Cham. https://doi.org/10.1007/978-3-319-07956-1_22

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  • DOI: https://doi.org/10.1007/978-3-319-07956-1_22

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07955-4

  • Online ISBN: 978-3-319-07956-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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