Chordality

C++ Program for detecting chordality. Chordal graphs are probably one of the most important objects in combinatorial commutative algebra. The algebraic importance of chordal graphs is twofolds. First, the independence complex of a chordal graph is shellable [5, Theorem 2.13] (even more, vertex decomposable [6, Corollary 5.5]) and hence the Stanley-Reisner ring of these complexes are sequentially Cohen-Macaulay. Second, chordal graphs lead to a beautiful classification of square-free monomial ideals with 2-linear resolution. Indeed, thanks to Fröberg, we know that a square-free monomial ideal I has a 2-linear resolution if and only if I is the edge ideal of a non-trivial graph G, such that the complement of G is a chordal graph. It turns out that, by using Alexander duality, the algebraic mechanism behind chordal graphs is quite rich. For example the following theorem explains the power of chordal graphs in combinatorial commutative algebra

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References in zbMATH (referenced in 1 article )

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  1. Bigdeli, Mina; Herzog, Jürgen; Yazdan Pour, Ali Akbar; Zaare-Nahandi, Rashid: Simplicial orders and chordality (2017)