(Almost) Complete intersection Lovász−Saks−Schrijver ideals and the regularity of their powers

We discuss the property of (almost) complete intersection of LSS-ideals of graphs of some special forms, like trees, unicyclic and bicyclic graphs. Further, we give a sufficient condition for the complete intersection property of twisted LSS-ideals in terms of a new graph theoretical invariant called twisted positive matching decomposition number denoted by $\mathrm{\text{tpmd}}$. We also study the regularity of powers of LSS-ideals and make observations related to the Koszul property of the quotients of the same.