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Characterization of partial 3-trees in terms of three structures

Abstract

A characterization of partial 3-trees is given based on the elimination sequence of vertices. It is proved that a partial 3-tree contains a vertex set satisfying either of certain three kinds of neighborhood relations on vertices and that a graph is a partial 3-tree if and only if eliminating such a vertex set results in a partial 3-tree. These results yield anO(n 2) time algorithm to recognize 3-trees.

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References

  1. 1.

    Beineke, L.W., Pippert, R.E.: Properties and characterizations ofk-trees. Mathematika18, 141–151 (1971)

  2. 2.

    Rose, D.J.: A graph theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Graph Theory and Computing, edited by R. Read. pp. 183–217. New York: Academic Press 1972

  3. 3.

    Rose, D.J.: On simple characterization ofk-trees. Discrete Math.7, 317–322 (1974)

  4. 4.

    Farley, A.M.: Networks immune to isolated failures. Networks11, 255–268 (1981)

  5. 5.

    Wald, J.A., Colbourn, C.J.: Steiner trees, partial 2-trees, and minimum IFI networks. Networks13, 159–167 (1983)

  6. 6.

    Proskurowski, A.: Separating subgraphs ink-trees; cables and caterpillars. Discrete Math.49, 275–285 (1984)

  7. 7.

    Knuth, D.E.: The art of computer programming; sorting and searching, vol. 3, 5.2, p. 170. Reading Mass. 1973

  8. 8.

    Ishizuka, A.: Master thesis. Tokyo Institute of Technology 1984

  9. 9.

    Kajitani, Y., Ishizuka, A., Ueno, S.: A characterization of the partial 3-tree in terms of certain structures. In: Proc. IEEE Int'l Symp. on Circuits and Systems (ISCAS 85), pp. 1179–1182. 1985

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Kajitani, Y., Ishizuka, A. & Ueno, S. Characterization of partial 3-trees in terms of three structures. Graphs and Combinatorics 2, 233–246 (1986). https://doi.org/10.1007/BF01788098

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Keywords

  • Time Algorithm
  • Neighborhood Relation
  • Elimination Sequence