Binomial edge ideals and conditional independence statements

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 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.

Algebraic properties of the binomial edge ideal of a complete bipartite graph

“…, y n . To the graph G one can associate an ideal J G ⊂ S generated by all binomials x i y j − x j y i for all 1 ≤ i < j ≤ n such that {i, j} is an edge of G. This construction was invented by Herzog et al in [6] and [4]. At first let us recall some of their definitions.…”

“…The binomial edge ideals of a graph G might have applications in algebraic statistics (see [6]). By the work of Herzog et al (see [6]) the Theorem 1.1. With the previous notation let J G ⊂ S denote the binomial edge ideal associated to the complete bipartite graph G = K m,n .…”

“…This is -at least for us -the first time in the literature that there is a complete description of the structure of the modules of deficiencies besides of sequentially Cohen-Macaulay rings or Buchsbaum rings. Our analysis is based on the primary decomposition of J G as shown in [6].…”

Decompositions of cellular binomial ideals

“…Combinatorially, binomial ideals already contain the class of monomial ideals, a cornerstone of combinatorial commutative algebra [13,18]. Moreover, equations with two terms are used in a variety of applied contexts where the primary decompositions of the corresponding binomial ideals carry important information (see for example [3,7,17,6] and the survey [12]). …”

The binomial ideal of the intersection axiom for conditional probabilities

“…A broad generalization of this paper's results to the class of binomial edge ideals of graphs has been obtained by Herzog, Hibi, Hreinsdóttir, Kahle, and Rauh [6]. The r = 2 case of I M is treated, with a different term order, in their Sect.…”