Precoloring extension has the usual graph coloring problem as a special case, in which the initially colored subset of vertices is empty; therefore, it is NP-complete.
For instance it is NP-complete on the rook's graphs, for which it corresponds to the problem of completing a partially filled-in Latin square.
[1] The problem may be solved in polynomial time for graphs of bounded treewidth, but the exponent of the polynomial depends on the treewidth.
[2][3] It may be solved in linear time for precoloring extension instances in which both the number of colors and the treewidth are bounded.
To transform a precoloring extension problem into a list coloring problem, assign each uncolored vertex in the precoloring extension problem a list of the colors not yet used by its initially-colored neighbors, and then remove the colored vertices from the graph.