  Let $G$ be a simple graph on $n$ vertices. Let $L_G \text{ and }
\mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal and parity
binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots,
x_n, y_1, \ldots, 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.
