max induced edges ex(w): {1: 0, 2: 1, 3: 2, 4: 4, 5: 5, 6: 7, 7: 9, 8: 12}
f(8) = 8a - c = 11 (must be <= 11)
w=2: a*w - c - ex(w) = 0 (must be >= 0)
w=3: a*w - c - ex(w) = 2/3 (must be >= 0)
w=4: a*w - c - ex(w) = 1/3 (must be >= 0)
w=5: a*w - c - ex(w) = 1 (must be >= 0)
w=6: a*w - c - ex(w) = 2/3 (must be >= 0)
w=7: a*w - c - ex(w) = 1/3 (must be >= 0)
merge of two clusters sharing s>=2 vertices: a*2 - c = 1 (must be >= 0)
several clusters, final case: a - c/8 = 11/8 ; 2(a - c/8) = 11/4 (must be >= a)
