Connectedness of digraphs from quadratic polynomials

Suppose that f(x)=x(x−k), where k is an odd positive integer. First, an infinite digraph G_k=(V,E) is defined, where the vertex set is V=ℤ and the edge set is E={(x,y)x,yZ,f(x)=f(2y)}. Then the following results are proved: If k=1, then the digraph G_k is weakly connected; if p is a safe prime, i.e., both p and q=(p−1)∕2 are primes, then the number w_p of weakly connected components of the digraph G_p is 2. Finally, a conjecture that there are infinitely many primes p such that w_p=2 is presented.