Details:
Title  Binomial edge ideals and conditional independence statements  Author(s)  Jürgen Herzog, Takayuki Hibi, Freyja Hreinsdottir, Johannes Rauh, Kahle Thomas  Type  Article in Journal  Abstract  We introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and clawfree. We give a reduced squarefree Gröbner basis for general G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation.  Keywords  Binomial ideals, Edge ideals, Cohen–Macaulay rings, Conditional independence ideals, Robustness  ISSN  01968858 
URL 
http://www.sciencedirect.com/science/article/pii/S019688581000014X 
Language  English  Journal  Advances in Applied Mathematics  Volume  45  Number  3  Pages  317  333  Year  2010  Edition  0  Translation 
No  Refereed 
No 
