Does distance-2 coloring fit in Telle and Proskurowski 's algorithm for partial-k trees?
distance-2 coloring is coloring in the square (d(x,y) &lt;= 2 implies xy an edge in the square). If a graph has tw k, its square has bounded clique-width (see Gurski-Wanke and Suchan-Todinca). See Oum et al. for algorithms as in Proskurowski-Telle for graphs of bounded clique-width.