Algorithms for the rainbow vertex coloring problem on graph classes

Paloma T Lima,Erik Jan Van Leeuwen,Marieke Van Der Wegen

doi:10.1016/j.tcs.2021.07.009

Paloma T Lima, Erik Jan Van Leeuwen + Show 1 more

Open Access

https://doi.org/10.1016/j.tcs.2021.07.009

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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.

Highlights

Graph coloring is a classic problem within the field of structural and algorithmic graph theory that has been widely studied in many variants
The Rainbow Vertex Coloring (RVC) problem takes as input a graph G and an integer k and asks whether G has a coloring with k colors under which it is rainbow vertex-connected
If G is a permutation graph on n vertices, rvc(G) = diam(G) − 1 and the corresponding rainbow vertex coloring can be found in O(n2) time

Summary

Introduction

Graph coloring is a classic problem within the field of structural and algorithmic graph theory that has been widely studied in many variants. The analogous computational problem is called Strong Rainbow Vertex Coloring (SRVC) and the corresponding parameter is denoted srvc(G) Both the RVC and the SRVC problems are NP-complete for every k ≥ 2 [4, 3, 5], and remain NP-complete even on bipartite graphs and split graphs [7]. If G is a permutation graph on n vertices, rvc(G) = diam(G) − 1 and the corresponding rainbow vertex coloring can be found in O(n2) time. This generalizes the earlier result on bipartite permutation graphs [7]. If G is a power of a tree, rvc(G) ∈ {diam(G) − 1, diam(G)}, and the corresponding optimal rainbow vertex coloring can be found in time that is linear in the size of G

Preliminaries

Permutation graphs

Split strongly chordal graphs

Powers of trees

Conclusion and open problems