Let $G$ be the graph with vertex set those pairs $(x,y)\in \mathbb{N}^2$ with $\mathrm{gcd}(x,y)=1$, in which we join two vertices if the differ in only one coordinate, and there by $\pm 1$.

Core of the composite-anchor reduction: vertical-leg vertices $(a, s)$ for $b \le s \le c$ are valid vertices of the strengthened problem, given the anchor $a$ is composite and coprime to the whole leg.