Fixing Improper Colorings of Graphs
Résumé
In this paper we consider a variation of a recoloring problem, called the r-Color-Fixing. Let us have some non-proper r-coloring $\varphi$ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is "the most similar" to $\varphi$, i.e. the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any r \geq 3, but is Fixed Parameter Tractable (FPT), when parametrized by the number of allowed transformations k. We provide an O∗(2^n) algorithm for the problem (for any fixed r) and a linear algorithm for graphs with bounded treewidth. Finally, we investigate the fixing number of a graph G. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of G and a proper one.