Is it obligatory to maintain ties with a sister who insists on staying with her swindler, con-artist husband, visiting him in prison, and giving him money and food, and is it permissible to give her zakat?

Maintaining kinship ties is one of the greatest acts of drawing closer to Allah. The sister is among the kin whose ties must be maintained. Her refusal to divorce her imprisoned husband does not justify abandoning her. As for giving her zakat (definition), if she is self-sufficient, it is not permissible. But if she is in need, you may give her zakat, and she has more right to your zakat, because charity given to a relative brings a doubled reward: the reward for charity and the reward for maintaining kinship ties.