Strong intractability results for generalized convex recoloring problems

A coloring of the vertices of a connected graph is r-convex if each color class induces a subgraph with at most r components. We address the r-convex recoloring problem defined as follows. Given a graph  G and a coloring of its vertices, recolor a minimum number of vertices of G so that the resultin...

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Bibliographic Details
Published in:Discrete Applied Mathematics 2020-07, Vol.281, p.252-260
Main Authors: Moura, Phablo F.S., Wakabayashi, Yoshiko
Format: Article
Language:English
Subjects:
Apexes
Bipartite graph
Coloring
Convex coloring
Convexity
Graph theory
Hardness
Inapproximability
Proteins
Trees (mathematics)
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Summary:A coloring of the vertices of a connected graph is r-convex if each color class induces a subgraph with at most r components. We address the r-convex recoloring problem defined as follows. Given a graph  G and a coloring of its vertices, recolor a minimum number of vertices of G so that the resulting coloring is r-convex. This problem, known to be NP-hard even on paths, was firstly investigated on trees and for r=1, motivated by applications on perfect phylogenies. The more general concept of r-convexity, for r≥2, was proposed later, and it is also of interest in the study of protein–protein interaction networks and phylogenetic networks. In this work, we show that, for each  r∈N, the r-convex recoloring problem on n-vertex bipartite graphs cannot be approximated within a factor of  n1−ε for any ε>0, unless P=NP. We also provide strong hardness results for weighted and parameterized versions of the problem.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2019.08.002