TY - GEN
T1 - Algorithms for the rainbow vertex coloring problem on graph classes
AU - Lima, Paloma T.
AU - van Leeuwen, Erik Jan
AU - van der Wegen, Marieke
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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.
AB - 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.
KW - Permutation graphs
KW - Powers of trees
KW - Rainbow vertex coloring
UR - http://www.scopus.com/inward/record.url?scp=85090510986&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.MFCS.2020.63
DO - 10.4230/LIPIcs.MFCS.2020.63
M3 - Conference contribution
AN - SCOPUS:85090510986
T3 - Leibniz International Proceedings in Informatics (LIPIcs)
SP - 63:1--63:13
BT - 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
A2 - Esparza, Javier
A2 - Kral, Daniel
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
Y2 - 25 August 2020 through 26 August 2020
ER -