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Interval k-Graphs and Orders

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Abstract

An interval k-graph is the intersection graph of a family of intervals of the real line partitioned into k classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we study the cocomparability interval k-graphs; that is, the interval k-graphs whose complements have a transitive orientation and are therefore the incomparability graphs of strict partial orders. For brevity we call these orders interval k-orders. We characterize the kind of interval representations a cocomparability interval k-graph must have, and identify the structure that guarantees an order is an interval k-order. The case k = 2 is peculiar: cocomparability interval 2-graphs (equivalently proper- or unit-interval bigraphs, bipartite permutation graphs, and complements of proper circular-arc graphs to name a few) have been characterized in many ways, but we show that analogous characterizations do not hold if k > 2. We characterize the cocomparability interval 3-graphs via one forbidden subgraph and hence interval 3-orders via one forbidden suborder.

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References

  1. Brandstädt, A., Spinrad, J., Stewart, L.: Bipartite permutation graphs. Discret. Appl. Math. 18, 279–292 (1987)

    Article  MathSciNet  Google Scholar 

  2. Brown, D.E., Flesch, B.M.: A characterization of 2-tree proper interval 3-graphs. J. Discret. Math. 2014, 143809 (2014). https://doi.org/10.1155/2014/143809

    Article  MATH  Google Scholar 

  3. Brown, D.E., Langley, L.J.: The Mathematics Of Preference, Choice and Order: Essays in Honor of Peter C. Fishburn, ch. Probe Interval Orders, pp 313–322. Springer, Heidelberg (2009)

    Book  Google Scholar 

  4. Brown, D.E.: Variations on Interval Graphs. Ph.D. thesis, University of Colorado Denver (2004)

  5. Brown, D.E., Lundgren, J.R.: Bipartite probe interval graphs, interval point bigraphs, and circular arc graphs. Aust. J. Commun. 35, 221–236 (2006)

    MATH  Google Scholar 

  6. Brown, D.E., Lundgren, J.R.: Characterizations for unit interval bigraphs. Congr. Numer. 206, 5–17 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Brown, D.E., Lundgren, J.R., Sheng, L.: Cycle-free unit and proper probe interval graphs, submitted to Discret. Appl. Math.

  8. Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal triple-free graphs. SIAM J. Discrete Math. 10, 399–430 (1997)

    Article  MathSciNet  Google Scholar 

  9. Das, S., Roy, A.B., Sen, M., West, D.B.: Interval digraphs: an analogue of interval graphs. Journal of Graph Theory 13(2), 189–202 (1989)

    Article  MathSciNet  Google Scholar 

  10. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)

    Article  MathSciNet  Google Scholar 

  11. Dushnik, B., Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941). MR MR0004862 (3,73a)

    Article  MathSciNet  Google Scholar 

  12. Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)

    Article  MathSciNet  Google Scholar 

  13. Gallai, T.: Transitiv orientbare graphen. Acta Math. Acad. Sci. Hungar 18, 25–66 (1967)

    Article  MathSciNet  Google Scholar 

  14. Golumbic, M.C., Rotem, D., Urrutia, J.: Comparability graphs and intersection graphs. Discret. Math. 43, 37–46 (1983)

    Article  MathSciNet  Google Scholar 

  15. Hayward, R.B.: Weakly triangulated graphs. J. Comb. Theory (B) 39, 200–209 (1985)

    Article  MathSciNet  Google Scholar 

  16. Hell, P., Huang, J.: Interval bigraphs and circular arc graphs. J. Graph Theory 46, 313–327 (2004)

    Article  MathSciNet  Google Scholar 

  17. Hiraguchi, T.: On the dimension of orders. Sci. Reports Kanazawa Univ. 4(4), 1–20 (1955)

    MathSciNet  MATH  Google Scholar 

  18. McKee, T., McMorris, F.R.: Topics in Intersection Graph Theory. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  Google Scholar 

  19. McMorris, F.R., Wang, C., Zhang, P.: On probe interval graphs. Discret. Appl. Math. 88, 315–324 (1998)

    Article  MathSciNet  Google Scholar 

  20. Sanyal, B.K., Sen, M.K.: Indifference digraphs: a generalization of indifference graphs and semiorders. SIAM J. Discrete Math. 7(2), 157–165 (1994)

    Article  MathSciNet  Google Scholar 

  21. Sheng, L.: Cycle-free probe interval graphs. Congressus Numerantium 88, 33–42 (1999)

    MathSciNet  MATH  Google Scholar 

  22. Spinrad, J.: Circular-arc graphs with clique cover number two. J. Comb. Theory, Series B 44(3), 300–306 (1987)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84341-3900, USA

    David E. Brown

  2. Mathematics Department, University of the Pacific, Stockton, CA, 95211, USA

    Larry J. Langley

  3. Computer Science Division, Western Oregon University, Monmouth, OR, 97361, USA

    Breeann M. Flesch

Authors
  1. David E. Brown
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  2. Breeann M. Flesch
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  3. Larry J. Langley
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Correspondence to David E. Brown.

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Brown, D.E., Flesch, B.M. & Langley, L.J. Interval k-Graphs and Orders. Order 35, 495–514 (2018). https://doi.org/10.1007/s11083-017-9445-0

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  • DOI: https://doi.org/10.1007/s11083-017-9445-0

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