References

Agnarsson, G. and R. Greenlaw, 2007. Graph Theory, Modeling, Applications and Algorithms. Pearson/Prentice Hall, Upper Saddle River, New Jersey, USA., ISBN:9780131423848, Pages: 446.

Aoki, S. and A. Takemura, 2003. Minimal basis for a connected Markov chain over 3×3×K contingency tables with fixed two‐dimensional marginals. Aust. N. Z. J. Stat., 45: 229-249.
CrossRef  |  Direct Link  |  

Aoki, S. and A. Takemura, 2003. The list of indispensable moves of the unique minimal markov basis of 3×4×K and 4×4×K contingency tables with fixed two-dimensional marginals. Master Thesis, Department of Mathematical Engineering and Information Physics, The University of Tokyo, Tokyo, Japan.

Aoki, S. and A. Takemura, 2008. The largest group of invariance for Markov bases and toric ideals. J. Symbolic Comput., 43: 342-358.
Direct Link  |  

Clark, J. and D.A. Holton, 1995. A First Look at Graph Theory. World Scientific Publishing, Singapore, ISBN:81-7023-463-8, Pages: 325.

Diaconis, P. and B. Sturmfels, 1998. Algebraic algorithms for sampling from conditional distributions. Ann. Stat., 26: 363-397.
Direct Link  |  

Diaconis, P., D. Eisenbud and B. Sturmfels, 1998. Lattice Walks and Primary Decomposition. In: Mathematical Essays in Honor of Gian-Carlo Rota, Sagan, B. and R. Stanley (Eds.). Birkhauser Basel, Basel, Switzerland, ISBN:978-1-4612-8656-1, pp: 173-193.

Dobra, A. and S. Sullivant, 2002. A divide-and-conquer algorithm for generating markov bases of multi-way tables. Proceedings of the Workshop on Multidimensional Tables Statistics, Combinatorial Optimization and Groebner Bases, November 17-20, 2002, University of California Davis, Davis, California, pp: 1-20.

Dobra, A., 2003. Markov bases for decomposable graphical models. Bernoulli, 9: 1093-1108.
CrossRef  |  Direct Link  |  

Russell, P., 2001. Genetics. Benjamin Cummings, New York, USA.,.

Sullivant, S., J. Rauh and T. Kahle, 2012. Postive margins and primary decomposition. Master Thesis, North Carolina State University, Raleigh, North Carolina.

Sullivant, S.M., 2005. Toric ideals in algebraic statistics. Ph.D Thesis, University of California, Berkeley, California.

Takemura, A. and S. Aoki, 2004. Some characterizations of minimal Markov basis for sampling from discrete conditional distributions. Annal. Inst. Stat. Math., 56: 1-17.
CrossRef  |  Direct Link  |  

Takemura, A. and S. Aoki, 2005. Distance-reducing Markov bases for sampling from a discrete sample space. Bernoulli, 11: 793-813.
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