When you are together as a couple with your children's other parent, you have a relatively large degree of control over the people your children are exposed to. However, after a divorce, or break up, it is typical for that control to decrease. It can be frustrating, as many parents new to this situation wonder if they can prevent their children from being in contact with certain people. Unfortunately, unless your ex agrees otherwise, the short answer is no, unless there is a significant risk of harm to the children from said exposure.
Why is this so? Shouldn't you, as a parent, have a right to say who your children should and should not be around? The answer is yes, but so does the other parent. For example, if your ex's mother is domineering, why can't you get a court order to block this woman from being around your children on your ex's time? Well, unless the children's grandmother presents some sort of potential danger or safety risk to the children, your ex is going to be able to decide who the children are going to be around on their time, just as you decide who the children are exposed to on your time.
The rationale behind this is that the law in W.Va. presumes that a fit parent acts in the children's best interest and that having a relationship with grandparents - even the domineering type - may be in the children's best interest. Your ex is going to get to make those decisions on their time, just as you make those decisions on your time. However, some courts may limit the parties from exposing the children to a new significant other during the initial split of the parents or during the divorce proceeding. For example, some judges may order that neither parent can have the children around any new partners for the first six months after the parties' split.
Also, this is not to say that a court will never enter an order prohibiting the children from being exposed to certain individuals, but again, there must be a potential safety concern to the children. Some examples where the court may grant the request would be a person who has a history of inappropriate discipline, significant criminal charges relating to children or individuals with a proven history of physical or sexual abuse, domestic violence, or individuals with substance abuse problems. However, even in these situations, the court may order that these people cannot be around the children unsupervised rather than prohibit all contact, which means that your ex or another competent individual would have to be present with the children if the potentially dangerous individual was present as well.