Subgraph induced by negative cycles detected by Bellman-Ford algorithm

What is known about the properties of the subgraph induced by the negative cycles are defined by the predecessor relation that is established during the execution of the Bellman-Ford algorithm and arcs $(u,v)$ satisfying $mathrm{dist}(u)+omega(u,v)ltmathrm{dist}(v)$ after its execution?

Assuming that the negative cycles are vertex-disjoint, is it true that re-running the Bellman-Ford algorithm after the weights of negative-cycle arcs have been multiplied by $-1$, no negative cycles will be found in that 2nd execution?