Acta Applicandae Mathematicae

, Volume 126, Issue 1, pp 245–252 | Cite as

Bi-resolving Graph Homomorphisms and Extensions of Bi-closing codes



Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We investigate the bi-covering extensions of bi-resolving homomorphisms and give several sufficient conditions for a bi-resolving homomorphism to have a bi-covering extension with an irreducible domain. Using these results, we prove that a bi-closing code between subshifts can be extended to an n-to-1 code between irreducible shifts of finite type for all large n.


Covering Resolving Subamalgamation matrix Covering extension Bi-closing Shift of finite type 

Mathematics Subject Classification (2000)

37B10 05C50 05C70 37B40 


  1. 1.
    Adler, R., Goodwyn, W., Weiss, B.: Equivalence of topological Markov shifts. Isr. J. Math. 27, 48–63 (1977) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adler, R., Marcus, B.: Topological entropy and equivalence of dynamical systems. Mem. Am. Math. Soc. 219 (1979) Google Scholar
  3. 3.
    Ashley, J.: An extension theorem for closing maps of shifts of finite type. Trans. Am. Math. Soc. 336, 389–420 (1993) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ashley, J., Marcus, B., Perrin, D., Tuncel, S.: Surjective extensions of sliding-block codes. SIAM J. Discrete Math. 6, 582–611 (1993) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math. 243, 21–66 (2002) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Boyle, M.: Lower entropy factors of sofic systems. Ergod. Theory Dyn. Syst. 3, 541–557 (1983) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Boyle, M.: Open problems in symbolic dynamics. In: Burns, K., Dolgopyat, D., Pezin, Y. (eds.) Geometric and Probabilistic Structures in Dynamics. Contemporary Mathematics Series, vol. 469, pp. 69–118. American Mathematical Society, Providence (2008) CrossRefGoogle Scholar
  8. 8.
    Boyle, M., Krieger, W.: Almost Markov and shift equivalent sofic systems. In: Lecture Notes in Math., vol. 1342. Springer, Berlin (1988) Google Scholar
  9. 9.
    Boyle, M., Tuncel, S.: Infinite-to-one codes and Markov measures. Trans. Am. Math. Soc. 285, 657–684 (1984) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Deng, A., Sato, I., Wu, Y.: Homomorphisms, representations and characteristic polynomials of digraphs. Linear Algebra Appl. 423, 386–407 (2007) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gross, J., Tucker, T.: Generating all graph coverings by permutation voltage assignments. Discrete Math. 18, 273–283 (1977) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gross, J., Tucker, T.: Topological Graph Theory. Wiley-Interscience, New York (1987) MATHGoogle Scholar
  13. 13.
    Jung, U.: On the existence of open and bi-continuing codes. Trans. Am. Math. Soc. 363, 1399–1417 (2011) MATHCrossRefGoogle Scholar
  14. 14.
    Kim, K.H., Roush, F.: On the structure of inert automorphisms of subshifts. Pure Math. Appl., Ser. B 2, 3–22 (1991) MathSciNetMATHGoogle Scholar
  15. 15.
    Kim, K.H., Roush, F.: Free Z p actions on subshifts. Pure Math. Appl. 8, 293–322 (1997) MathSciNetMATHGoogle Scholar
  16. 16.
    Kitchens, B.: Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts. Springer, Berlin (1998) MATHGoogle Scholar
  17. 17.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995) MATHCrossRefGoogle Scholar
  18. 18.
    Nasu, M.: Constant-to-one and onto global maps of homomorphisms between strongly connected graphs. Ergod. Theory Dyn. Syst. 3, 387–413 (1983) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Trow, P.: Degrees of finite-to-one factor maps. Isr. J. Math. 71, 229–238 (1990) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    van Lint, J., Wilson, R.: A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2001) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsAjou UniversitySuwonSouth Korea
  2. 2.The Jesuit Novitiate of St. Stanislaus in KoreaSuwonSouth Korea

Personalised recommendations