, Volume 81, Issue 11–12, pp 4258–4274 | Cite as

Hardness and Structural Results for Half-Squares of Restricted Tree Convex Bipartite Graphs

  • Hoang-Oanh Le
  • Van Bang LeEmail author
Part of the following topical collections:
  1. Special Issue: Computing and Combinatorics


Let \(B=(X,Y,E)\) be a bipartite graph. A half-square of B has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y. Every planar graph is a half-square of a planar bipartite graph, namely of its subdivision. Until recently, only half-squares of planar bipartite graphs, also known as map graphs (Chen et al., in: Proceedings of the thirtieth annual ACM symposium on the theory of computing, STOC ’98, pp 514–523., 1998; J ACM 49(2):127–138., 2002), have been investigated, and the most discussed problem is whether it is possible to recognize these graphs faster and simpler than Thorup’s \(O(n^{120})\)-time algorithm (Thorup, in: Proceedings of the 39th IEEE symposium on foundations of computer science (FOCS), pp 396–405., 1998). In this paper, we identify the first hardness case, namely that deciding if a graph is a half-square of a balanced bisplit graph is NP-complete. (Balanced bisplit graphs form a proper subclass of star convex bipartite graphs). For classical subclasses of tree convex bipartite graphs such as biconvex, convex, and chordal bipartite graphs, we give good structural characterizations of their half-squares that imply efficient recognition algorithms. As a by-product, we obtain new characterizations of unit interval graphs, interval graphs, and of strongly chordal graphs in terms of half-squares of biconvex bipartite, convex bipartite, and of chordal bipartite graphs, respectively. Good characterizations of half-squares of star convex and star biconvex bipartite graphs are also given, giving linear-time recognition algorithms for these half-squares.


Half-square NP-hardness Graph algorithm Computational complexity Graph classes 

Mathematics Subject Classification

68R10 05C85 68Q25 



We thank Hannes Steffenhagen for his careful reading and very helpful remarks. We also thank one of the unknown referees for his/her helpful comments and suggestions, and for pointing out a gap in an earlier proof of Lemma 4.


  1. 1.
    Brandenburg, F.J.: On 4-map graphs and 1-planar graphs and their recognition problem. CoRR (2015). arXiv:1509.03447
  2. 2.
    Chen, Z.-Z.: Approximation algorithms for independent sets in map graphs. J. Algorithms 41, 20–40 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z.-Z., Grigni, M., Papadimitriou, C.H.: Planar map graphs. In: Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, STOC ’98, pp. 514–523 (1998).
  4. 4.
    Chen, Z.-Z., Grigni, M., Papadimitriou, C.H.: Map graphs. J. ACM 49(2), 127–138 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Z.-Z., He, X., Kao, M.-Y.: Nonplanar topological inference and political-map graphs. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 195–204 (1999)Google Scholar
  6. 6.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Fixed-parameter algorithms for \((k, r)\)-center in planar graphs and map graphs. ACM Trans. Algorithms 1(1), 33–47 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Demaine, E.D., Hajiaghayi, M.T.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008). CrossRefGoogle Scholar
  8. 8.
    Eickmeyer, K., Kawarabayashi, K.-I.: FO model checking on map graphs. In: Fundamentals of Computation Theory—21st International Symposium, FCT 2017, pp. 204–216. Bordeaux, France (2017). CrossRefGoogle Scholar
  9. 9.
    Farber, M.: Characterizations of strongly chordal graphs. Discrete Math. 43, 173–189 (1983). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fomin, F.V., Lokshtanov, D., Saurabh, S.: Bidimensionality and geometric graphs. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 1563–1575 (2012).
  11. 11.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  14. 14.
    Heggernes, P., Kratsch, D.: Linear-time certifying recognition algorithms and forbidden induced subgraphs. Nord. J. Comput. 14, 87–108 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Holyer, I.: The NP-completeness of some edge-partition problems. SIAM J. Comput. 4, 713–717 (1981). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jiang, W., Liu, T., Wang, C., Xu, K.: Feedback vertex sets on restricted bipartite graphs. Theor. Comput. Sci. 507, 41–51 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kou, L.T., Stockmeyer, L.J., Wong, C.-K.: Covering edges by cliques with regard to keyword conflicts and intersection graphs. Commun. ACM 21, 135–139 (1978). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lau, L.C.: Bipartite roots of graphs. ACM Trans. Algorithms 2(2), 178–208 (2006). (Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA (2004) pp. 952–961.)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lau, L.C., Corneil, D.G.: Recognizing powers of proper interval, split, and chordal graphs. SIAM J. Discrete Math. 18(1), 83–102 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Le, H.-O., Le, V.B.: Hardness and structural results for half-squares of restricted tree convex bipartite graphs. In: Proceedings of the 23rd Annual International Computing and Combinatorics Conference, COCOON, pp. 359–370 (2017). Google Scholar
  21. 21.
    Le, V.B., Peng, S.-L.: On the complete width and edge clique cover problems. In: Proceedings of the 21st Annual International Computing and Combinatorics Conference 2015, COCOON, pp. 537–547 (2015). Extended version to appear in Journal ofCombinatorial Optimization, available on-line under Google Scholar
  22. 22.
    Lehel, J.: A characterization of totally balanced hypergraphs. Discrete Math. 57, 59–65 (1985). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liu, T.: Restricted bipartite graphs: comparison and hardness results. In: Proceedings of the Tenth International Conference on Algorithmic Aspects of Information and Management, pp. 241–252 . LNCS 8546, AAIM, (2014)Google Scholar
  24. 24.
    Lubiw, A.: Doubly lexical orderings of matrices. SIAM J. Comput. 16, 854–879 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    McConnell, R.M.: Linear-time recognition of circular-arc graphs. Algorithmica 37, 93–147 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefGoogle Scholar
  27. 27.
    Mnich, M., Rutter, I., Schmidt, J.M.: Linear-time recognition of map graphs with outerplanar witness. In: Proceedings of the 15th Scandinavian Symposium and Workshops on Algorithm Theory, Article No. 5, SWAT, pp. 5:1–5:14 (2016).
  28. 28.
    Motwani, R., Sudan, M.: Computing roots of graphs is hard. Discrete Appl. Math. 54(1), 81–88 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Orlin, J.: Contentment in graph theory: covering graphs with cliques. Indag. Math. 80(5), 406–424 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic Press, Cambridge (1969)Google Scholar
  31. 31.
    Spinrad, J.P.: Efficient Graph Representations. Fields Institute Monographs, Toronto (2003)CrossRefGoogle Scholar
  32. 32.
    Thorup, M.: Map graphs in polynomial time. In: Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, FOCS, pp. 396–405 (1998).

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.BerlinGermany
  2. 2.Institut für InformatikUniversität RostockRostockGermany

Personalised recommendations