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TitleBinomial edge ideals and conditional independence statements
Author(s) Jürgen Herzog, Takayuki Hibi, Freyja Hreinsdottir, Johannes Rauh, Kahle Thomas
TypeArticle in Journal
AbstractWe 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 claw-free. 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.
KeywordsBinomial ideals, Edge ideals, Cohen–Macaulay rings, Conditional independence ideals, Robustness
URL http://www.sciencedirect.com/science/article/pii/S019688581000014X
JournalAdvances in Applied Mathematics
Pages317 - 333
Translation No
Refereed No