Fixing improper colorings of graphs

Abstract

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper r-coloring ϕ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is "the most similar" to ϕ, 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 fixed r ≥ 3, even for bipartite planar graphs. Moreover, it is W [1]-hard even for bipartite graphs, when parameterized by the number k of allowed recoloring transformations. On the other hand, the problem is fixed-parameter tractable, when parameterized by k and the number r of colors.We provide a 2 n ⋅n O(1) algorithm for the problem and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the fixing number of a graph G. It is the minimum k such that k recoloring transformations are sufficient to transform any coloring of G into a proper one.

On the complexity of restoring corrupted colorings

“…By relaxing the requirement on the promise condition, we establish again the following. We also note that using different ideas, the same conclusion was reached in [13].…”

“…• In Section 4, we prove that under plausible complexity assumptions, r-Fix has no polynomial kernel parameterized by the number of recolorings k, for every r ≥ 3. We stress that while mentioned as an open problem in [20], the question was subsequently answered by Garnero, Junosza-Szaniawski, Liedloff, Montealegre, and Rzążewski in a full version [13] of [20]. Our result was obtained independently of their work, and uses slightly different ideas.…”

“…Thus, they asked whether or not there is a kernel of polynomial size. The question was answered in the negative in a full version of [20] by Garnero et al [13]. Independently of their work, in what is to follow, we give an alternative proof of the theorem.…”

“…For a discussion on several related reoptimization and reconfiguration problems, we refer the reader to [20]. We also note that their results are considerably expanded in a full version of the manuscript currently under review [13].…”

Introduction to Reconfiguration

“…In one case, the graphs are directed, and the goal is to reach a particular proper coloring by giving a vertex a color different from that of its neighbors [25]. In COLOR-FIXING, starting with an assignment of colors to vertices, the goal is to determine the minimum number of vertex recolorings required in order to find any proper coloring [26].…”