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Representing triangulated graphs in stars

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Abstract

A graphG is calledrepresentable in a tree T, ifG is isomorphic to the intersection graph of a family of subtrees ofT. In this paper those graphs are characterized which are representable in some subdivision of theK 1,n. In the finite case polynomial-time recognition algorithms of these graphs are given. But this concept can be generalized to essentially infinite graphs by using no more trees but ‘tree-like’ posets and representability of graphs in these posets.

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References

  1. [1]

    P.A. Bernstein, N. Goodmann, The Power of Natural Semijoins, SIAM J. Computing10 (1981) 751–771.

  2. [2]

    K.S. Booth, G.S. Leuker, Testing for the Consecutive Ones *-Property, Interval Graphs, and Graph Planarity Using PQ-tree Algorithms, J. Comput. Syst. Sci.13 (1976) 335–379.

  3. [3]

    R.P. Dilworth, A Decomposition Theorem for Partially Ordered Sets, Annals of Mathematics51 (1950) 161–166.

  4. [4]

    D.R. Fulkerson, Note on Dilworth’s Decomposition Theorem for Partially Ordered Sets, Proc. Amer. Math. Soc.7 (1956) 701–702.

  5. [5]

    F. Gavril, The Intersection Graphs of Subtrees in Trees are Exactly the Chordal Graphs, J. Comb. Th.B 16 (1974) 47–56.

  6. [6]

    R. Halin, Some Remarks on Interval Graphs, Combinatorica2 (1982).

  7. [7]

    R. Halin, Graphentheorie II, Wissenschaftliche Buchgesellschaft Darmstadt 1981.

  8. [8]

    J.E. Hopcroft, R.M. Karp, Ann 5/2 Algorithm for Maximum Matchings in Bipartite Graphs, SIAM J. Comput.2 (1973) 225–231.

  9. [9]

    C. Lekkerkerker, J. Boland, Representation of a Finite Graph by a Set of Intervals on the Real Line, Fund. Math.51 (1962) 45–64.

  10. [10]

    E. Prisner, Tree Representation of Chordal Graphs and the Weighted Clique Graph, unpublished manuscript 1986.

  11. [11]

    D.J. Rose, R.E. Tarjan, G.S. Leuker, Algorithmic Aspects of Vertex Elimination on Graphs, SIAM J. Comput.5 (1976) 266–283.

  12. [12]

    Y. Shibata, On the Tree-representation of Chordal Graphs, J. Graph Th.12 (1988) 421–428.

  13. [13]

    J.R. Walter, Representations of Chordal Graphs as Subtrees of a Tree, J. Graph Th.2 (1978) 265–267.

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Correspondence to E. Prisner.

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Prisner, E. Representing triangulated graphs in stars. Abh.Math.Semin.Univ.Hambg. 62, 29–41 (1992). https://doi.org/10.1007/BF02941616

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Keywords

  • Intersection Graph
  • Interval Graph
  • Cardinal Number
  • Chordal Graph
  • Star Graph