Lovász–Saks–Schrijver ideals and parity binomial edge ideals of graphs

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