Green’s Dictionary of Slang

b.d.v. n.

[abbr. bend down Virginia]

(UK tramp) a cigarette stub, picked up in the street.

[UK]M. Marshall Tramp-Royal on the Toby 154: These are the shagsters, or hard-up men. Their blend [...] is called hard-up, kerbstone twist, B.D.V. (Bend Down Virginia), and other names.