Abstract
A graph is chordal if and only if it has no chordless cycle of length more than three. The set of maximal cliques in a chordal graph admits special tree structures called clique trees. A perfect sequence is a sequence of maximal cliques obtained by using the reverse order of repeatedly removing the leaves of a clique tree. This paper addresses the problem of enumerating all the perfect sequences. Although this problem has statistical applications, no efficient algorithm has been proposed. There are two difficulties with developing this type of algorithms. First, a chordal graph does not generally have a unique clique tree. Second, a perfect sequence can normally be generated by two or more distinct clique trees. Thus it is hard using a straightforward way to generate perfect sequences from each possible clique tree. In this paper, we propose a method to enumerate perfect sequences without constructing clique trees. As a result, we have developed the first polynomial delay algorithm for dealing with this problem. In particular, the time complexity of the algorithm on average is O(1) for each perfect sequence.
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References
Blair, J.R.S., Peyton, B.: An Introduction to Chordal Graphs and Clique Trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation. IMA, vol. 56, pp. 1–29. Springer, Heidelberg (1993)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)
Chandran, L.S., Ibarra, L., Ruskey, F., Sawada, J.: Generating and Characterizing the Perfect Elimination Orderings of a Chordal Graph. Theoretical Computer Science 307, 303–317 (2003)
Dahl, J., Vandenberghe, L., Roychowdhury, V.: Covariance Selection for Non-chordal Graphs via Chordal Embedding. Optimization Methods and Software 23(4), 501–520 (2008)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Annals of Discrete Mathematics 57. Elsevier, Amsterdam (2004)
Hara, H., Takemura, A.: Boundary Cliques, Clique Trees and Perfect Sequences of Maximal Cliques of a Chordal Graph. METR 2006-41, Department of Mathematical Informatics, University of Tokyo (2006)
Hara, H., Takemura, A.: Bayes Admissible Estimation of the Means in Poisson Decomposable Graphical Models. Journal of Statistical Planning and Inference (in press, 2008)
Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs. SIAM Journal on Computing 28(5), 1906–1922 (1999)
Lauritzen, S.L.: Graphical Models. Clarendon Press, Oxford (1996)
McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM, Philadelphia (1999)
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic Aspects of Vertex Elimination on Graphs. SIAM Journal on Computing 5(2), 266–283 (1976)
Spinrad, J.P.: Efficient Graph Representations. American Mathematical Society (2003)
Tarjan, R.E., Yannakakis, M.: Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs. SIAM Journal on Computing 13(3), 566–579 (1984)
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Matsui, Y., Uehara, R., Uno, T. (2008). Enumeration of Perfect Sequences of Chordal Graph. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_75
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DOI: https://doi.org/10.1007/978-3-540-92182-0_75
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