  We study monomial cut ideals associated to a graph $G$, which are a monomial
analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary
decompositions, projective dimensions, and Castelnuovo-Mumford regularities are
investigated if the graph can be decomposed as $0$-clique sums and disjoint
union of subgraphs. The total Betti numbers of a cycle are computed. Moreover,
we classify all Freiman ideals among monomial cut ideals.
