Given the problem 1/m-Clique (half clique third clique and so on) ,I wanted to know if the following reduction is correct, since every solution I've seen in the internet was far more complicated and most of the times used the reduction from 1/(m-1) Clique problem, instead of doing it directly from the Clique problem. So I want to show a pol' reduction from Clique to 1/m-Clique for some m. Given (G,k) ,build G': take G and add (n-k) vertices and connect them to every node in G (so if there's an k clique, now there will be an n clique), and add m*n-(2n-k) isolated vertices, return (G',n). overall if there's a k clique in G, there will be an n clique in G', and m*n vertices, hence n/m*n=1/m clique. If G doesn't have a k clique (nor j>k clique) no n-clique will be constructed in G' (we only add n-k nodes which can "help" in forming a clique), hence no 1/m clique.
1/m Clique reduction