Let $G$ be a simple graph on $n$ vertices. Let $L_G$ and ${\mathcal{I}}_G$ denote the Lovász–Saks–Schrijver(LSS) ideal and parity binomial edge ideal of $G$ in the polynomial ring $S=\mathbb{K}[x_1,\dots ,x_n,y_1,\dots ,y_n]$ respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen–Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs.

让 $G$ 成为一个简单的图 $ñ$顶点。让${\mathrm{大号}}_G$ 和 ${\mathcal{一世}}_G$ 表示Lovász-Saks-Schrijver（LSS）理想和奇偶二项式边理想 $G$ 在多项式环中 $\mathrm{小号}=\mathbb{ķ}[X_{\mathrm{1个}}，\dots ，X_{ñ}，{ÿ}_{\mathrm{1个}}，\dots ，{ÿ}_{ñ}]$分别。我们对图的LSS理想和奇偶二项式边缘理想是完全相交的图进行分类。我们还对图的LSS理想和奇偶二项式边缘理想几乎是完整的交集进行分类，并且证明了它们的Rees代数是Cohen–Macaulay。我们计算第二级贝蒂数，并获得树木和奇数单圈图的LSS理想值的最小表示。我们还获得了树和奇单环图的LSS理想的对称代数的定义理想的明确描述。