Locally Injective Homomorphism to the Simple Weight Graphs

  • Ondřej Bílka
  • Bernard Lidický
  • Marek Tesař
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)


A Weight graph is a connected (multi)graph with two vertices u and v of degree at least three and other vertices of degree two. Moreover, if any of these two vertices is removed, the remaining graph contains a cycle. A Weight graph is called simple if the degree of u and v is three. We show full computational complexity characterization of the problem of deciding the existence of a locally injective homomorphism from an input graph G to any fixed simple Weight graph by identifying some polynomial cases and some NP-complete cases.


computational complexity locally injective homomorphism Weight graph 


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  1. 1.
    Abello, J., Fellows, M.R., Stillwell, J.C.: On the complexity and combinatorics of covering finite complexes. Australian Journal of Combinatorics 4, 103–112 (1991)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: The classification of coverings of processor networks. Journal of Parallel Distributed Computing 6, 166–182 (1989)CrossRefGoogle Scholar
  3. 3.
    Feder, T., Vardi, M.Y.: Monotone monadic SNP and constraint satisfaction. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC 1993, pp. 612–622 (1993)Google Scholar
  4. 4.
    Fiala, J.: NP completeness of the edge precoloring extension problem on bipartite graphs. Journal of Graph Theory 43, 156–160 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fiala, J.: Locally injective homomorphisms, disertation thesis (2000)Google Scholar
  6. 6.
    Fiala, J., Kratochvíl, J.: Complexity of partial covers of graphs. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 537–549. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discussiones Mathematicae Graph Theory 22, 89–99 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fiala, J., Kratochvíl, J.: Locally injective graph homomorphism: Lists guarantee dichotomy. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 15–26. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Fiala, J., Kratochvíl, J., Pór, A.: On the computational complexity of partial covers of Theta graphs. Discrete Applied Mathematics 156, 1143–1149 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fiala, J., Paulusma, D.: The computational complexity of the role assignment problem. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 817–828. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Hell, P., Nešetřil, J.: On the complexity of H-colouring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering regular graphs. Journal of Combinatorial Theory B 71, 1–16 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering directed multigraphs I. colored directed multigraphs. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 242–257. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  14. 14.
    Kratochvíl, J., Proskurowski, A., Telle, J.A.: Complexity of graph covering problems. Nordic Journal of Computing 5, 173–195 (1998)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kristiansen, P., Telle, J.A.: Generalized H-coloring of graphs. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 456–466. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  16. 16.
    Lidický, B., Tesař, M.: Locally injective homomorphism to the Theta graphs. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 326–336. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC 1978, pp. 216–226 (1978)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ondřej Bílka
    • 1
  • Bernard Lidický
    • 1
  • Marek Tesař
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPragueCzech Republic

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