Parts of speech

[Lostinlang]

I'm trying to explain the parts of speech involved in different English sentences to a non-native speaker. She came up with this one, which has me stumped; 'A dog-walking method' similar to 'my shower-cleaning method'. I'm flailing around between adjectives, adverbs and genitive case. Any help appreciated.

[Alan]

'Dog-walking' is essentially a compound adjective here, modifying the noun 'method'.