Fixing improper colorings of graphs

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 $r$-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 $r \geq 3$, even for bipartite planar graphs. On the other hand, the problem is fixed-parameter tractable, when parameterized by the number of allowed transformations $k$. We provide an $2^n \cdot n^{\mathcal{O}(1)}$ algorithm for the problem (for any fixed $r$) 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 $G$. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of $G$ and a proper one.