The bottom line is that she has created a scenario where she has to trespass on your property in order to maintain her 1 ft of leftover yard on your side of her fence. I think that is a good idea to fence off backyard completely so that she has to wait for you to unlock and let her in and she can clean her 1 ft while you watch. Or you can fence it off and tell her you will clean it yourself. Build your raised beds 1 ft away from the fence and then line the 1 ft with plastic and put down a thick layer of gravel or mulch. Nothing will grow there and neither you or her will have to maintain anything there. In Missouri, btw, It is my understanding that that 1 ft of property will end up belonging to you! I do think if she has a problem with them, she cannot do anything about it. You do not have to allow her onto your yard in order to maintain hers. You have that right. It is a problem SHE created for herself, so basically she just forfeited a foot of her yard.
