Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes

Authors Paloma T. Lima, Erik Jan van Leeuwen, Marieke van der Wegen

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Paloma T. Lima
  • Department of Informatics, University of Bergen, Norway
Erik Jan van Leeuwen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Marieke van der Wegen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
  • Mathematical Institute, Utrecht University, The Netherlands

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Paloma T. Lima, Erik Jan van Leeuwen, and Marieke van der Wegen. Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed p ≥ 3 both variants of the problem become NP-complete when restricted to split (S₃,…,S_p)-free graphs, where S_q denotes the q-sun graph.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • rainbow vertex coloring
  • permutation graphs
  • powers of trees


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