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Determining what sets of trees can be the clique trees of a chordal graph
Journal of the Brazilian Computer Society volume 18, pages 121–128 (2012)
Abstract
Chordal graphs have characteristic tree representations, the clique trees. The problems of finding one or enumerating them have already been solved in a satisfactory way. In this paper, the following related problem is studied: given a family of trees, all having the same vertex set V, determine whether there exists a chordal graph whose set of clique trees equals
. For that purpose, we undertake a study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree. Some necessary and sufficient conditions and examples of how they can be applied are found, eventually establishing that a positive or negative answer to the problem can be obtained in polynomial time. If affirmative, a graph whose set of clique trees equals
is also obtained. Finally, all the chordal graphs with set of clique trees equal to
are characterized.
References
De Caria P, Gutierrez M (2011) Comparing trees characteristic to chordal and dually chordal graphs. Electron Notes Discrete Math 37:33–38
Dirac GA (1961) On rigid circuit graphs. Abh Math Semin Univ Hamb 25:71–76
Galinier P, Habib M, Paul C (1995) Chordal graphs and their clique graphs. Lect Notes Comput Sci 1017:358–371
Gavril F (1974) The intersection graphs of subtrees in trees are exactly the chordal graphs. J Comb Theory, Ser B 116:47–56
Gavril F (1987) Generating the maximum weight spanning trees of a weighted graph. J Algorithms 8:592–597
Golumbic MC (2004) Algorithmic graph theory and perfect graphs, 2nd edn. Annals of discrete mathematics, vol 57. Elsevier, Amsterdam, p 10
Habib H, Stacho J (2011) Reduced clique graphs of chordal graphs. Accepted to Eur J Combin
Kruskal JB (1956) On the shortest spanning tree of a graph and the traveling salesman problem. Proc Am Math Soc 7:48–50
Bernstein PA, Goodman N (1981) Power of natural semijoins. SIAM J Comput 10:751–771
Shibata Y (1988) On the tree representation of chordal graphs. J Graph Theory 12:421–428
Sreenivasa Kumara P, Veni Madhavanb CE (2002) Clique tree generalization and new subclasses of chordal graphs. Discrete Appl Math 117:109–131
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De Caria, P., Gutierrez, M. Determining what sets of trees can be the clique trees of a chordal graph. J Braz Comput Soc 18, 121–128 (2012). https://doi.org/10.1007/s13173-011-0048-0
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DOI: https://doi.org/10.1007/s13173-011-0048-0