Complexity of Rainbow Vertex Connectivity Problems for Restricted Graph Classes

TL;DR

This paper investigates the computational complexity of rainbow vertex connectivity problems in various restricted graph classes, establishing NP-completeness in many cases and identifying classes where the problems are tractable.

Contribution

It precisely characterizes the complexity of rainbow vertex connectivity problems across multiple graph classes and parameters, including NP-completeness and fixed-parameter tractability results.

Findings

NP-complete on bipartite planar graphs of max degree 3

NP-complete on interval graphs and k-regular graphs for k ≥ 3

Polynomial-time solvable on block, cactus, and split graphs

Abstract

A path in a vertex-colored graph $G$ is \emph{vertex rainbow} if all of its internal vertices have a distinct color. The graph $G$ is said to be \emph{rainbow vertex connected} if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph $G$ is \emph{strongly rainbow vertex connected} if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the complexity of deciding if a given vertex-colored graph is rainbow or strongly rainbow vertex connected. We call these problems \probRvc and \probSrvc, respectively. We prove both problems remain NP-complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and $k$-regular graphs for $k \geq 3$. We settle precisely the complexity of both problems from the viewpoint of two width parameters: pathwidth and tree-depth. More precisely, we show both problems remain NP-complete for bounded pathwidth graphs, while being fixed-parameter tractable parameterized by tree-depth. Moreover, we show both problems are solvable in polynomial time for block graphs, while \probSrvc is tractable for cactus graphs and split graphs.