Homeowners and Gardeners, Can You Help? What IS This?

I have never seen anything like this. Our neighbors had been watering their yard and taking care of it (seemingly) until one day I saw this:

I swore I thought they just sprayed something wrong on it, or something. This is the side of their house, along the driveway. The hose is ours.

Each day it seems to get worse, too, and because they are our NEXT DOOR neighbors, it’s now beginning to seep into our yard, too (the end of the fence ends their yard):

Here’s a couple close-ups. What I thought was dying grass actually seems to be vines of some sorts, or roots.

The folks next door don’t own this house, so they aren’t necessarily going to care if they are killing the landlord’s grass. But we DO own this house, it was brand new sod last year, and we ARE keeping care of it. What can we do!?

Please, someone, anyone, HELP!