Precoloring extension
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In graph theory, precoloring extension is the problem of extending a graph coloring of a subset of the vertices of a graph, with a given set of colors, to a coloring of the whole graph that does not assign the same color to any two adjacent vertices.
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. However, it is also NP-complete for some other classes of graphs on which the usual graph coloring problem is easier. 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.[2]