stair-carpet vs rug

stair-carpet and rug both are nouns.

Word	Noun	Adjective	Verb	Adverb
stair-carpet	Yes	No	No	No
rug	Yes	No	No	No

As nouns, rug is a hypernym of stair-carpet; that is, rug is a word with a broader meaning than stair-carpet:

stair-carpet: a strip of carpet for laying on stairs
rug: floor covering consisting of a piece of thick heavy fabric (usually with nap or pile)

Other hypernyms of stair-carpet include carpet, carpeting.

stair-carpet (noun)	rug (noun)
a strip of carpet for laying on stairs	floor covering consisting of a piece of thick heavy fabric (usually with nap or pile)