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Restricted Bipartite Graphs: Comparison and Hardness Results

  • Tian Liu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8546)

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.

Keywords

Perfect elimination bipartite graphs tree convex bipartite graphs circular convex bipartite graphs chordal bipartite graphs convex bipartite graphs \(\mathcal{NP}\)-completeness treewidth feedback vertex set 

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References

  1. 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)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Brandstad, A., Le, V.B., Spinrad, J.P.: Graph Classes - A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)Google Scholar
  3. 3.
    Golumbic, M.C., Goss, C.F.: Perfect elimination and chordal bipartite graphs. J. Graph Theory 2, 155–163 (1978)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Grover, F.: Maximum matching in a convex bipartite graph. Nav. Res. Logist. Q. 14, 313–316 (1967)CrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. 6.
    Jiang, W., Liu, T., Wang, C., Xu, K.: Feedback vertex sets on restricted bipartite graphs. Theor. Comput. Sci. 507, 41–51 (2013)CrossRefMATHMathSciNetGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    Karp, R.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  9. 9.
    Kloks, T.: Treewidth: Computations and Approximations. Springer (1994)Google Scholar
  10. 10.
    Kloks, T., Kratsch, D.: Treewidth of chordal bipartite graphs. J. Algorithms 19, 266–281 (1995)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kloks, T., Liu, C.H., Pon, S.H.: Feedback vertex set on chordal bipartite graphs. arXiv:1104.3915 (2011)Google Scholar
  12. 12.
    Kloks, T., Wang, Y.L.: Advances in Graph Algorithms (2013) (manuscript)Google Scholar
  13. 13.
    Liang, Y.D., Blum, N.: Circular convex bipartite graphs: Maximum matching and Hamiltonian circuits. Inf. Process. Lett. 56, 215–219 (1995)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Liu, T., Lu, Z., Xu, K.: Tractable connected domination for restricted bipartite graphs. J. Comb. Optim. (2014), , doi 10.1007/s10878-014-9729-xGoogle Scholar
  15. 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)CrossRefGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 20.
    Tucker, A.: A structure theorem for the consecutive 1’s property. Journal of Combinatorial Theory 12(B), 153–162 (1972)Google Scholar
  21. 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. 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)CrossRefGoogle Scholar
  23. 23.
    Yannakakis, M.: Node-deletion problem on bipartite graphs. SIAM J. Comput. 10, 310–327 (1981)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Tian Liu
    • 1
  1. 1.Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina

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