Complexity of virtual multistrings

A virtual $n$-string $\alpha$ is a collection of $n$ oriented smooth generic loops on a surface $M$. A stabilization of $\alpha$ is a surgery that results in attaching a handle to $M$ along disks avoiding $\alpha$, and the inverse operation is a destabilization of $\alpha$. We consider virtual $n$-strings up to virtual homotopy, i.e., sequences of stabilizations, destabilizations, and homotopies of $\alpha$. Recently, Cahn proved that any virtual $1$-string can be virtually homotoped to a genus-minimal and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of connected non-parallel $n$-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual $1$-string are related by Type 3 moves, stabilizations, and destabilizations. Kadokami claimed that this held for virtual $n$-strings in general, but Gibson found a counterexample for $5$-strings. We show that Kadokami's statement holds for connected non-parallel $n$-strings and exhibit a counterexample for $3$-strings.