Green’s Dictionary of Slang

do-me-dag n.

[rhy. sl. = fag n.3 (2); ult. do my dags, a children’s game synon. with ‘follow my leader’; the image is of children encouraging each other to smoke]

a cigarette.

[UK]N&Q 12 Ser. IX 346: Do-Me-Dags. Fags; cigarettes.
[UK]Partridge DSUE (8th edn) 318/2: late C.19–20.
[UK]R. Puxley Fresh Rabbit.