The d-precoloring problem for k-degenerate graphs

In this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXT problem
is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at
most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.

We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted
to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted
to the class of partial k-trees (without any size restriction on S).We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use
for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and
its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.