Fixing improper colorings of graphs
Valentin Garnero, Konstanty Junosza-Szaniawski, Mathieu Liedloff, and Pedro Montealegre, Pawe{\l} Rz\k{a}\.zewski

TL;DR
This paper studies the problem of minimally recoloring a graph to fix an improper coloring, showing NP-completeness, fixed-parameter tractability, and providing algorithms and complexity bounds.
Contribution
It introduces the color-fixing problem, proves its NP-completeness for r ≥ 3, and offers fixed-parameter algorithms and bounds for graphs with bounded treewidth.
Findings
NP-complete for r ≥ 3, even for bipartite planar graphs
Fixed-parameter tractable when parameterized by the number of recolorings
Provides algorithms and complexity bounds for the problem
Abstract
In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper -coloring of a graph . We investigate the problem of finding a proper -coloring of , which is "the most similar" to , i.e. the number of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any , even for bipartite planar graphs. On the other hand, the problem is fixed-parameter tractable, when parameterized by the number of allowed transformations . We provide an algorithm for the problem (for any fixed ) and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the {\em fixing number} of a graph . It is the maximum possible…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
