On powers of graphs of bounded NLC-width (clique-width)

Given a graph G  , the graph GlGl has the same vertex set and two vertices are adjacent in GlGl if and only if they are at distance at most l in G. The l  -coloring problem consists in finding an optimal vertex coloring of the graph GlGl, where G is the input graph. We show that, for any fixed value of l, the l  -coloring problem is polynomial when restricted to graphs of bounded NLC-width (or clique-width), if an expression of the graph is also part of the input. We also prove that the NLC-width of GlGl is at most 2(l+1)nlcw(G)2(l+1)nlcw(G).