A few things to try
Tell them what the pup is doing and just express concern, don't make them think you are being an butt head about it and they should be responsive.
The chicken wire is a great idea or cinder block or buried bricks.
You could also do a hot wire so if the dog did dig under they would get corrected.
:goodpost: !!!
take a garden hoe; after you've raised the awarness of the neighbors if they don't already know

and go right next to the fence, make a trench with the garden hoe about the exact size of the blade, depth and width wise, roughly 3 in deep and 5 in wide... go get you some quickcrete which is specialized premixed concrete fast drying for multiple uses.. .. ONLY WAY TO STOP DIGGING ON A YARD WITH PRIVACY OR WOODEN FENCE(not yelling expressing concern, been there done that, even in rent home your doing everybody a favor.
NOW if you have chain link, take rebar from welding shop or steel yard, where rebar is expendable and scrapped like framers do with lumber.. Take the rebar and slide it intertwinded through the bottom of the chainlink that touches the ground, usually a tension wire is put there but its not APBT proof,

the rebar is.
To fill the holes, once again quickcrete; I used the dog poo method myself, filled the holes with dog

and then covered with gravel and then dirt..
It worked and it didnt.. Started crappin on the fence line where they ran as oppose to their designated spot out of the way in the yard, so then I had dog turds flippin up off there feet, cause they

where they smell it needs to be

up:
I've always had this problem and when in rent homes or building my kennel, I used cement/concrete (quickcrete)
and rebar
... and large rocks for filler in the deep holes..
make your cement go as far as possible us 3/4 crushed gravel w/ sand.. some bags come pre mixed, so its up to what you can find and what works best for you. Both can be found at Home Depot or Lowes...
JFYI DONT TOUCH THE FENCE IF ITS WOODEN AND DAMAGED FROM THE OTHER DOG, TAKE PICs and show your landlord and the neighbor so its clear its not your dog. It is the cost of both neighbors generally to get a fence on the the exact property line, and its the neighbors responcibility to fix sections of damaged fence; If you have to get the cedar planks to have him replace the chewed or destroyed fencing, the cedar planks are more cost effective than making your yard look like Sandford&Son with plywood, sheetmetal, and chicken wire; thats money waisted in my experience. Just replace and repair it right the first time and make sure its APBT proof.. AGAIN if the other dog has caused damages to the cedar planks that needs to be fixed first, keep all receipts and take before and after pics, take pics of the dog destroying the fence and diggin under; if the neighbors see you doing this, they will fix the fence or yell at you for them being ignorant.. just my experiences.
If you do it that way; like an algebra equation, this way will work... There are other ways that work, but this way is guaranteed
