Algorithms for the rainbow vertex coloring problem on graph classes

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Abstract

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, . . ., Sp)-free graphs, where Sq denotes the q-sun graph.

Original languageEnglish
Title of host publication45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
EditorsJavier Esparza, Daniel Kral
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages63:1--63:13
Number of pages13
ISBN (Electronic)9783959771597
DOIs
Publication statusPublished - 1 Aug 2020
Event45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020 - Prague, Czech Republic
Duration: 25 Aug 202026 Aug 2020

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
Volume170
ISSN (Print)1868-8969

Conference

Conference45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
Country/TerritoryCzech Republic
CityPrague
Period25/08/2026/08/20

Keywords

  • Permutation graphs
  • Powers of trees
  • Rainbow vertex coloring

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