Following the success of the previous question:
Karp hardness of an equidistant vertex set
I continue to propse yet another computational problem. This time, we modify the notion of an equidistant vertex set to obtain the notion of a simply equidistant vertex set. A vertex set $V'\subseteq V$ is called simply equidistant if there exists some $\alpha\in \mathbb{N}$ such that for every $u, v\in V'$, there exists a simple path of length $\alpha$ connecting $u$, $v$.
Our problem:
Input: an undirected graph $G(V,E)$ and a natural number $k$
Output: YES if there exists a simply equidistant of size $k$, otherwise NO
What is the hardness of this problem?
