The complexity of changing colourings with bounded maximum degree

Let G be a graph, x,y∈V(G), and ϕ:V(G)→[k] a k-colouring of G such that ϕ(x)=ϕ(y). If then the following question is NP-complete: Does there exist a k-colouring ϕ′ of G such that ϕ′(x)≠ϕ′(y)? Conversely, if then the problem is polynomial time.