Advertisement

A Comprehensive Introduction to the Theory of Word-Representable Graphs

  • Sergey KitaevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

Abstract

Letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word \(xyxy\cdots \) (of even or odd length) or a word \(yxyx\cdots \) (of even or odd length). A graph \(G=(V,E)\) is word-representable if and only if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if \(xy\in E\).

Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. This paper offers a comprehensive introduction to the theory of word-represent-able graphs including the most recent developments in the area.

References

  1. 1.
    Akrobotu, P., Kitaev, S., Masárová, Z.: On word-representability of polyomino triangulations. Siberian Adv. Math. 25(1), 1–10 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, T.Z.Q., Kitaev, S., Sun, B.Y.: Word-representability of face subdivisions of triangular grid graphs. Graphs Comb. 32(5), 1749–1761 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, T.Z.Q., Kitaev, S., Sun, B.Y.: Word-representability of triangulations of grid-covered cylinder graphs. Discr. Appl. Math. 213(C), 60–70 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Collins, A., Kitaev, S., Lozin, V.: New results on word-representable graphs. Discr. Appl. Math. 216, 136–141 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gao, A., Kitaev, S., Zhang, P.: On 132-representable graphs. arXiv:1602.08965 (2016)
  6. 6.
    Glen, M.: Colourability and word-representability of near-triangulations. arXiv:1605.01688 (2016)
  7. 7.
  8. 8.
    Glen, M., Kitaev, S.: Word-representability of triangulations of rectangular polyomino with a single domino tile. J. Comb. Math. Comb. Comput. (to appear)Google Scholar
  9. 9.
    Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Graphs capturing alternations in words. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 436–437. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14455-4_41 CrossRefGoogle Scholar
  10. 10.
    Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Alternation graphs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 191–202. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-25870-1_18 CrossRefGoogle Scholar
  11. 11.
    Halldórsson, M., Kitaev, S., Pyatkin, A.: Semi-transitive orientations and word-representable graphs. Discrete Appl. Math. 201, 164–171 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jones, M., Kitaev, S., Pyatkin, A., Remmel, J.: Representing graphs via pattern avoiding words. Electron. J. Comb. 22(2), 2.53 (2015). 20 ppMathSciNetzbMATHGoogle Scholar
  13. 13.
    Kim, J., Kim, M.: Graph orientations on word-representable graphs (in preparation)Google Scholar
  14. 14.
    Kitaev, S.: Patterns in Permutations and Words. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kitaev, S.: On graphs with representation number 3. J. Autom. Lang. Comb. 18(2), 97–112 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kitaev, S.: Existence of \(u\)-representation of graphs. J. Graph Theor. 85(3), 661–668 (2017)CrossRefGoogle Scholar
  17. 17.
    Kitaev, S., Lozin, V.: Words and Graphs. Springer, Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kitaev, S., Pyatkin, A.: On representable graphs. J. Autom. Lang. Comb. 13(1), 45–54 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kitaev, S., Salimov, P., Severs, C., Úlfarsson, H.: On the representability of line graphs. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 478–479. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22321-1_46 CrossRefGoogle Scholar
  20. 20.
    Kitaev, S., Seif, S.: Word problem of the Perkins semigroup via directed acyclic graphs. Order 25(3), 177–194 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mandelshtam, Y.: On graphs representable by pattern-avoiding words. arXiv:1608.07614 (2016)
  22. 22.
    Pretzel, O.: On graphs that can be oriented as diagrams of ordered sets. Order 2(1), 25–40 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Thomassen, C.: A short list color proof of Grötzsch’s theorem. J. Comb. Theor. Ser. B 88(1), 189–192 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer and Information SciencesUniversity of StrathclydeGlasgowUK

Personalised recommendations