A strongly independent set of a graph $G'$ is a subset $S$ of vertices such that the distance in $G'$ between every two distinct vertices in $S$ is larger than $2$.
It is possible to reduce an instance $\langle G, k\rangle$ of the decision version of independent set to an instance $\langle G', k\rangle$ of the decision version of strongly interdependent set by (1) replacing each edge $e=(u,v)$ of $G$ with two edges $(u, w_e)$ and $(w_e, v)$, where $w_e$ is a new vertex; and (2) connecting all the vertices $w_e$ with a clique.
In the newly created Graph $G'$, why does adding a middle node prevent us from choosing that node? Basically, how does the reduction step help us?
Solution Notes: http://www.cse.iitd.ernet.in/~amitk/SemI-2015/tut11.pdf