Complexity of virtual multistrings

A virtual $n$-string $\alpha$ is a collection of $n$ closed curves on an oriented surface $M$. Associated to $\alpha$, there are two natural measures of complexity: the genus of $M$ and the number of intersection points. By considering virtual $n$-strings up to equivalence by virtual homotopy, i.e., homotopies of the component curves and stabilizations/destabilizations of the surface, a natural question is whether these quantities can be minimized simultaneously. We show that this is possible for non-parallel virtual $n$-strings and that, moreover, such a representative can be obtained by monotonically decreasing genus and the number of intersections from any initial representative.