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Convexity and Discrete Geometry Including Graph Theory pp 27–35Cite as

Transformations of Digraphs Viewed as Intersection Digraphs

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 148))

Abstract

The intersection number of a digraph D is the minimum size of a set U, such that D is the intersection digraph of ordered pairs of subsets of U. The paper describes much of the work done in the area of intersection graphs and digraphs, and proves two main results: Theorem 1 The intersection number of the line digraph of D equals the number of vertices of D that are neither sources nor sinks. Theorem 2 If D contains no loops, the intersection numbers of total digraph, middle digraph and subdivision digraph of D are all equal to the number of vertices of D that are not sources, added to the number of vertices of D that are not sinks.

Support for Christina Zamfirescu’s work was provided by PSC-CUNY Awards, jointly funded by The Professional Staff Congress and The City University of New York.

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References

  1. J. Akiyama, T. Hamada, I. Yoshimura, On characterizations of the middle graphs. TRU Math. 11, 35–39 (1975)

    MathSciNet  MATH  Google Scholar 

  2. M. Behzad, A criterium for the planarity of the total graph. Proc. Cambridge Philos. Soc. 63, 679–681 (1967)

    Google Scholar 

  3. L.W. Beineke, Characterization of derived graphs. J. Comb. Theory B 9, 129–135 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. L.W. Beineke, C.M. Zamfirescu, Connection digraphs and second order line digraphs. Discrete Math 39, 237–254 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. J.C. Bermond, J.M.S. Simões-Pereira, C. Zamfirescu, On non-hamiltonian homogeneously traceable digraphs. Math. Japan 24, 423–426 (1979)

    MathSciNet  MATH  Google Scholar 

  6. S. Bertz, W.F. Wright: The graph theory approach to synthetic analysis: definition and application of molecular complexity and synthetic complexity, Graph Theory Notes of New York XXXV (1998) 32–18

    Google Scholar 

  7. S. Bertz, C. Zamfirescu, New complexity indices based on edge covers. Match 42, 39–70 (2000)

    MathSciNet  MATH  Google Scholar 

  8. P. Buneman, A characterization of rigid circuit graphs. Disc. Math. 9, 205–212 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. G. Chartrand, M.J. Stewart, Total digraphs. Canadian Math. Bull 9, 171–176 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Chikodimath, Sampathkumar: Semi-total graphs-II. Graph Theory Research Report, Karnath University No. 2, 5–9 (1973)

    Google Scholar 

  11. J.E. Cohen, Food Webs and Niche Space (Princeton University Press, Princeton, 1978)

    Google Scholar 

  12. P. Erdős, A. Goodman, L. Pósa, The representation of a graph by set intersections. Can. J. Math 18, 106–112 (1966)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory B 16, 47–56 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. P.C. Gilmore, A.J. Hoffman, A characterization of comparability graphs and the interval graphs. Canadian J. Math 16, 539–548 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.R. Griggs, D.B. West, Extremal values of the interval number of a graph. SIAM J. Alg. Disc. Math. 1, 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.R. Griggs, Extremal values of the interval number of a graph, II. Disc. Math. 28, 37–47 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Hadwiger, H. DeBrunner, V. Klee, Combinatorial Geometry in the Plane (Holt, Rinehart and Winston, New York, 1964)

    Google Scholar 

  19. G. Hajos: Über eine Art von Graphen: Intern. Math Nachr. 2, 65 (1957)

    Google Scholar 

  20. T. Hamada, I. Yoshimura, Traversability and connectivity of the middle graph of a graph. Discr. Math 14, 247–255 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Hanlon: Counting interval graphs, J. London Math Soc (1979)

    Google Scholar 

  22. F. Harary, Graph Theory (Addison-Wesley, Addison-Wesley Series in Mathematics, 1969)

    MATH  Google Scholar 

  23. F. Harary, J.A. Kabell: Infinite-interval graphs, Calcutta Mathematical Society. Diamond- cum-Platinum Jubilee Commemoration Volume (1908–1983) Part I, Calcutta Math. Soc., Calcutta (1984) 27–31

    Google Scholar 

  24. F. Harary, R.Z. Norman, Some properties of line-digraphs. Rend. Circ. Mat. Palermo 9, 161–168 (1960)

    Google Scholar 

  25. C. Heuchenne: Sur une certaine correspondence entre graphes, Bull. Soc. Roy. Liège 33 (1964) 743–753

    Google Scholar 

  26. D.G. Kendall, Incidence matrices, interval graphs and seriation in archaeology. Pacific J. Math 28, 565–570 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. V. Klee, What are the intersection graphs of arcs in a circle. Amer. Math. Monthly 76, 810–813 (1969)

    Article  MathSciNet  Google Scholar 

  28. J. Krausz, Démonstration nouvelle d’un théorème de Whitney sur les réseaux. Mat. Fiz. Lapok 50, 75–89 (1943)

    Google Scholar 

  29. C.B. Lekkerkerker, J.C. Boland, Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)

    MathSciNet  MATH  Google Scholar 

  30. E. Marczewski, Sur deux propriétés des classes d’ensembles. Fund. Math 33, 303–307 (1945)

    Google Scholar 

  31. T.A. McKee, F.R. Mc Morris: Topics in Intersection Graph Theory, SIAM Monographs on Discr. Math and Applications, Philadelphia, 1999

    Google Scholar 

  32. F.S. Roberts: On the boxicity and the cubicity of a graph, Recent Progress in Combinatorics (W.T. Tutte, ed.) Academic Press, New York 1969, 301–310

    Google Scholar 

  33. F.S. Roberts, Discrete Mathematical Models (Prentice Hall, Englewood Cliffs, NJ, 1976)

    Google Scholar 

  34. H.S. van Rooij, Wilf: The interchange graphs of a finite graph, Acta Math. Acad, Sci Hungar. 16, 263–269 (1965)

    Google Scholar 

  35. M. Sen, S. Das, A.B. Roy, D.B. West, Interval digraphs - An analogue of interval graphs. J. Graph Th. 13, 189–202 (1989)

    Google Scholar 

  36. M. Skowronska, M.M. Syslo, C. Zamfirescu, An Algorithmic Characterization of Total Digraphs. J. Algorithms 7, 120–133 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. F.W. Stahl: Circular genetic maps, J. Cell Physiology 70 Sup. 1 (1967) 1–12

    Google Scholar 

  38. K.E. Stoffers, Scheduling of traffic lights - a new approach. Transportation Research 2, 199–234 (1968)

    Article  Google Scholar 

  39. W.T. Trotter Jr., F. Harary, On double and multiple interval graphs. J. Graph Theory 3, 205–211 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Whitney, Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150–168 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  41. C. Zamfirescu, Cyclic and cliquewise connectedness of line graphs. Discrete Math 170, 293–297 (1997)

    Google Scholar 

  42. C.M. Zamfirescu, Local and global characterizations of middle digraphs, Theory and Applications of Graphs (G. Chartrand, ed.) J. Wiley, New York (1981) 595–607

    Google Scholar 

  43. C.M. Zamfirescu, E. Celenti, D. Celenti, Transformation of Digraphs and Intersection Digraphs: Connections and Results, Graph Theory Notes of New York Academy of Sciences XLIII, 39–47 (2002)

    Google Scholar 

  44. C.M. Zamfirescu, E.S. Klein, Transformed digraphs represented as intersection digraphs - a uniform way of expression and search for optimality. Congressus Numerantium 110, 137–144 (1995)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Hunter College and Graduate Center, City University of New York, New York, NY, 10065–10036, USA

    Christina M. D. Zamfirescu

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  1. Christina M. D. Zamfirescu
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Correspondence to Christina M. D. Zamfirescu .

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Editors and Affiliations

  1. Einstein Institute for Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Karim Adiprasito

  2. Hungarian Academy of Sciences, Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. Mathematics of the Roumanian Academy, ``Simion Stoilow'' Institute of, Bucharest, Romania

    Costin Vilcu

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Zamfirescu, C.M.D. (2016). Transformations of Digraphs Viewed as Intersection Digraphs. In: Adiprasito, K., Bárány, I., Vilcu, C. (eds) Convexity and Discrete Geometry Including Graph Theory. Springer Proceedings in Mathematics & Statistics, vol 148. Springer, Cham. https://doi.org/10.1007/978-3-319-28186-5_2

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