This vine had original been planted next to the fence by the neigbour.  Apparently they didn't like it and poisoned all of it on their side and it came over and established itself firmly on my side.

I've found that mowing it does keep it back a bit, but it is persistent.  The book recommendations are basically the same as the others -- cut down to stems and apply direct glyphosate or triclopyr and 2,4-D (crossbow) directly on freshly cut exposed ends.

http://mdc.mo.gov/sites/default/files/resources/2012/05/japanesehoneysuckle.pdf