Minimum Average Distance Clique Trees

Chordal graphs have been extensively studied and have applications in various fields, including computational biology, sparse matrix computation, and graphical models. They are characterized by the existence of clique trees, whose vertices correspond to the maximal cliques of a chordal graph. In many applications, it is the clique tree of the chordal graph that is of greatest utility. In general, the number of clique trees can grow exponentially with the size of the chordal graph, and in some applications, particular clique trees have greater utility; we want additional criteria to select the most useful clique tree(s). A natural criterion in phylogenetics (and perhaps elsewhere) is that of compactness. In this paper, we formalize this criterion as the average distance between nodes, and present a characterization of clique trees that satisfies this criterion. We also develop a polynomial-time algorithm to find such a clique tree, and show that any minimum average-distance clique tree of a chordal graph c...

[1]  Ko-Wei Lih.  Rank inequalities for chordal graphs , 1993, Discret. Math..

[2]  Jan Karel Lenstra,et al.  The complexity of the network design problem , 1978, Networks.

[3]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[4]  I. Gutman,et al.  Wiener Index of Hexagonal Systems , 2002 .

[5]  D. Hartl,et al.  A primer of population genetics , 1981 .

[6]  James R. Walter,et al.  Representations of chordal graphs as subtrees of a tree , 1978, J. Graph Theory.

[7]  Richard C. T. Lee,et al.  Counting Clique Trees and Computing Perfect Elimination Schemes in Parallel , 1989, Inf. Process. Lett..

[8]  Adnan Darwiche,et al.  Modeling and Reasoning with Bayesian Networks , 2009 .

[9]  Pinar Heggernes,et al.  Sequential and parallel triangulating algorithms for Elimination Game and new insights on Minimum Degree , 2008, Theor. Comput. Sci..

[10]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[11]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[12]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[13]  D. Rose.  Triangulated graphs and the elimination process , 1970 .

[14]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

[15]  Philip A. Bernstein,et al.  Power of Natural Semijoins , 1981, SIAM J. Comput..

[16]  Peter Buneman,et al.  A characterisation of rigid circuit graphs , 1974, Discret. Math..

[17]  Dan Gusfield,et al.  The Multi-State Perfect Phylogeny Problem with Missing and Removable Data: Solutions via Integer-Programming and Chordal Graph Theory , 2009, RECOMB.

[18]  Frank Harary,et al.  Distance in graphs , 1990 .

[19]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[20]  David Fernández-Baca,et al.  The Perfect Phylogeny Problem , 2001 .

[21]  M. Steel.  The complexity of reconstructing trees from qualitative characters and subtrees , 1992 .

[22]  Charles Semple,et al.  Identifying phylogenetic trees , 2005, Discret. Math..

[23]  Pinar Heggernes,et al.  A wide-range algorithm for minimal triangulation from an arbitrary ordering , 2006, J. Algorithms.

[24]  G. Dirac.  On rigid circuit graphs , 1961 .

[25]  Dan Gusfield,et al.  Generalizing the Splits Equivalence Theorem and Four Gamete Condition: Perfect Phylogeny on Three-State Characters , 2009, SIAM J. Discret. Math..

[26]  Richard C. T. Lee,et al.  Efficient Parallel Algorithms for Finding Maximal Cliques, Clique Trees, and Minimum Coloring on Chordal Graphs , 1988, Inf. Process. Lett..

[27]  Charles Semple,et al.  A characterization for a set of partial partitions to define an X-tree , 2002, Discret. Math..

[28]  Christopher A. Meacham,et al.  Theoretical and Computational Considerations of the Compatibility of Qualitative Taxonomic Characters , 1983 .

[29]  Fanica Gavril,et al.  Generating the Maximum Spanning Trees of a Weighted Graph , 1987, J. Algorithms.

[30]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[31]  Martin S. Andersen,et al.  Chordal Graphs and Semidefinite Optimization , 2015, Found. Trends Optim..

[32]  Jeff A. Bilmes,et al.  Creating non-minimal triangulations for use in inference in mixed stochastic/deterministic graphical models , 2011, Machine Learning.

[33]  Barry W. Peyton,et al.  On Finding Minimum-Diameter Clique Trees , 1994, Nord. J. Comput..

[34]  Anne Berry,et al.  A simple algorithm to generate the minimal separators and the maximal cliques of a chordal graph , 2011, Inf. Process. Lett..