  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 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.
