What is the minimum number of clades necessary to partition Vertebrates while maintaining the classical groups of Birds, Mammals, and Amphibians?
Phylogenetic taxonomy of the Vertebrates (mun.ca) has the following graphic (14 clades total, minimum of 12 collapsing Amphibia):



Here, what was preivously 'Fish' (the paraphyletic group of non-tetrapod vertebrates) becomes 6 clades, and 'Reptiles' (the paraphyletic  group of non-avian Sauropsids) becomes 3.

It seems the question is equivalent to asking "what is the minimum number of clades necessary to partition Vertebrates while maintaining 'birds' as a clade?".