I try to prove the equivalence between the two following properties of an NP-Complete problem $P$:
(A) $P$ is unary NP-Complete if it is NP-Complete even if we encode the integers of the inputs with unary encoding.
(B) $P$ is strongly NP-Complete if there exists a constant $c \in \mathbb{N}$ such that $P$ is NP-Complete when we restrict $P$ to instances $x$ where the maximum integer $\max(x)$ of $x$ is lower than $|x|^c$.
In Garey, M. R., & Johnson, D. S. (1978). ``strong''np-completeness results: Motivation, examples, and implications., where the second definition is given, unary encoding is mentioned at the end and it is suggested the two are equivalent. However, I cannot find a formal proof of that equivalence. Maybe I missed something obvious and important.
(B) $\Rightarrow$ (A) is easy as, for every instance $x$ with $\max(x) \leq |x|^c$, we can encode every integer of $x$ with unary encoding in polynomial time.
I can not find a proof for (A) $\Rightarrow$ (B). Is there any reference with this proof or a simple argument ?
(Also proving the equivalence between the weak NP-Completeness (i.e. the existence of a pseudopolynomial algorithm) and the fact that the problem is polynomial when the integers are unary encoded is also simple but this does not prove that (A) $\Leftrightarrow$ (B).)
Thanks in advance
