Hi Elizabeth-- I'm so sorry to hear about your bamboo invasion. Bamboo is a very difficult invasive plant to contain and eliminate. Here is a link with some information on removing it, including what to do when it comes from a neighbor's property. Anything on your own property is your right to eliminate, but it will continue to creep into your yard without cooperation and permission from your neighbor to go further than your property line. 

That link will also provide you with information on removal of your current chunk of bamboo, installing a barrier to prevent more from coming in, and how to dispose of what you remove.

Good luck and let us know if we can help further!