# Karp hardness of a simply equidistant vertex set

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?