The d-precoloring problem for k-degenerate graphs

Abstract

In this paper we deal with the d-P RECOLORING E XTENSION ( d-P RE XT) problem in various classes of graphs. The d-P RE XT problem is the special case of P RECOLORING E XTENSION 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-P RE XT, 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-P RE XT: given an instance of the d-P RE XT 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.