Parsimonious Multigraphs

We view a loopless multigraph as a complete graph with nonnegative integer edge weights indicating the multiplicity of each edge. A multigraph realization of a given degree sequence is parsimonious if it has the least number of positive weight edges. We characterize the graphs that can appear as components in a parsimonious multigraph and show that minimizing the number of positive weight edges is equivalent to maximizing the number of cycle-free components.