A Gröbner basis characterization for chordal comparability graphs

Abstract

In this paper, we study toric ideals associated with multichains of posets. It is shown that the comparability graph of a poset is chordal if and only if there exists a quadratic Gröbner basis of the toric ideal of the poset. Strong perfect elimination orderings of strongly chordal graphs play an important role.

As árvores características dos grafos cordais comparabilidade não possuem grau limitado

“…3) A prova de que o limite superior acima é justo, no sentido de que existe um grafo cordal comparabilidade cuja dimensão linear é igual a 4 [Kierstead et al 1992]. 4) Uma caracterizac ¸ão algébrica dos grafos cordais comparabilidade, baseada no fato que um grafo de comparabilidade G R é cordal se e somente se as multicadeias de R de tamanho pelo menos 3 podem ser associadas, de uma maneira específica, a um ideal tórico cuja base de Gröbner é quadrática [Ohsugi and Hibi 2016].…”

The number of edges of the edge polytope of a finite simple graph

“…Remark 2.5. Let P be a finite partially ordered set and Com(P ) its comparability graph [6]. Then Com(P ) is perfect [1] and the stable set polytope of Com(P ) is equal to the chain polytope [7] of P .…”

On toric ideals arising from signed graphs

“…A (homogeneous) toric ideal not only defines a projective toric variety (see [6,24]), but also provides wide applications in other areas, such as algebraic statistics, dynamical system, hypergeometric differential equations, toric geometry, and graph theory, see [5,14,25]. Toric ideals arising from various kinds of combinatorial objects have been widely studied by many researchers, see [12,13,19,20] for some recent results. Particularly, the toric ideal of a graph or a digraph, which is the toric ideal associated with its vertex-edge incidence matrix, has been an interesting topic (see [2][3][4]7,8,17,18,21,22]).…”