Just a quick word here ,I have only skipped over the hello and suggestions on this thread so apologise if this has been said before.
Each time you cut a bit off it gives it a growth spurt so beware it could be getting stronger. My advice would be to spray with what has been suggested .a dosing discretely of course !
I'm not cutting any of it back, what's missing is from when the neighbour hacked at it
They barely leave the house, but any spray that has accidentally gone over the fence hasn't been noticed or commented on, nor has the sudden demise of their plant.
I wish I could hire myself out to get rid of pesky plants, it's so engrossing to see the plant wither and die. At the end of the day i'm just so glad that my fence is going to be safe.
Also in case anyone was wondering, the neighbours haven't commented on their plant's demise, nor have they done anything to rectify the problem themselves. In fact I haven't even seen them in the garden, except for pegging out washing. Oh and the tyres are still there.
One final question, will the plant regrow if they don't remove the dead parts? Or will it just sit there like a dead thing, looking messy but not actually doing any harm
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Haha Woodgreen Wonderboy, that's exactly how I look each time I see more dead leaves
Just a quick word here ,I have only skipped over the hello and suggestions on this thread so apologise if this has been said before.
Each time you cut a bit off it gives it a growth spurt so beware it could be getting stronger. My advice would be to spray with what has been suggested .a dosing discretely of course !
Do they go out much ?
I'm not cutting any of it back, what's missing is from when the neighbour hacked at it
They barely leave the house, but any spray that has accidentally gone over the fence
hasn't been noticed or commented on, nor has the sudden demise of their plant. 
By the sound of it your getting t o the "in control stage"!
I hope so Cangrandmafixiit, it does seem to be a huge improvement
Very well done,!
I wish I could hire myself out to get rid of pesky plants, it's so engrossing to see the plant wither and die.
At the end of the day i'm just so glad that my fence is going to be safe.
Also in case anyone was wondering, the neighbours haven't commented on their plant's demise, nor have they done anything to rectify the problem themselves. In fact I haven't even seen them in the garden, except for pegging out washing. Oh and the tyres are still there.
One final question, will the plant regrow if they don't remove the dead parts? Or will it just sit there like a dead thing, looking messy but not actually doing any harm
Fifty shades freed
The dead plant will stay looking sorry for itself
And you will sit happy knowing you have done a good job done !
You not seen your neighbours, maybe they got too close to the spray??