Help to explain the article about Split Contraction

[Alex12]

Hello all.

Recently I have read the article: 'A. Agrawa, D. Lokshtanov, S. Saurabh and M. Zehavi. Split Contraction: The Untold Story'.

I am interested in the theorem from the article. The theorem statement can be stated as: «SRBPC parameterized by the number of parts in R is W[1]-hard».

I did not understand what is Split Contraction, SRBPC, Perfect Code and W[1]-hard. Could you please explain it easier for me?

And I need explanation the proof of the theorem.

As I understood, the theorem proofs through some Lemmas. And I ask you to explain for me these lemmas easier.

Thanks.