<human>: My mom lives in a rent house that shares a fence with another rent house next door.  My mom has two dogs in the back yard, a Blue Heeler and a Corgi.  A few months back, a tree in the neighbors yard fell and damaged the fence.  It's in tact enough to keep my mom's larger dogs in, but broken in some places where tiny dogs could get through into her back yard.  On multiple occasions she has talked to her landlord and the neighbors about the fence; and the other house's landlord has also been notified that the fence needs to be fixed.  Since it was the neighbor's tree that damaged the fence, supposedly it was worked out that the neighbor or their landlord would fix the fence (which they never did).

That brings us current, where the neighbors left their little toy dog (maybe 3-4 pound dog) unsupervised in their back yard.  It came through the fence and was killed by the blue heeler.  My mom's dog has never tried to escape the fence, he's never shown aggression towards people or other dogs other than barking at people who walk through the front yard.  He plays fine with my mom's other dog and my dog and my sister's dogs when we visit.  It's just that this dog was way too small to be playing with him.

The neighbors want my mom to put her dog down.  It is our opinion that her dog has done nothing wrong, and while it's an unfortunate accident, the neighbors should have done a better job of supervising their dog when they knew it could get through the fence.  My mom is scared that it will cost her a bunch of money if they try to take her to court to force her destroy her dog.  We've had the dog for 8 years and he's been a loyal, obedient dog the whole time.  It would break our hearts if we had to put it down.

Anyone gone through something like this or have any suggestions?

Can you give me the gist of the text in a nutshell?
<bot>: Neighbor's little dog came through a hole in the fence that was damaged by tree and went unfixed by them and got himself killed.  Neighbors want my mom to put her dog down.