Abstract
In this paper we consider a variation of a recoloring problem, called the r-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 r ≥ 3, but is Fixed Parameter Tractable (FPT), when parametrized by the number of allowed transformations k. We provide an \(\mathcal{O}^*(2^n)\) 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.
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Junosza-Szaniawski, K., Liedloff, M., Rzążewski, P. (2015). Fixing Improper Colorings of Graphs. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, JJ., Wattenhofer, R. (eds) SOFSEM 2015: Theory and Practice of Computer Science. SOFSEM 2015. Lecture Notes in Computer Science, vol 8939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46078-8_22
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DOI: https://doi.org/10.1007/978-3-662-46078-8_22
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