Further hardness results on the rainbow vertex-connection number of graphs

TL;DR

This paper proves that determining whether the rainbow vertex-connection number of a graph is at most a fixed integer k is an NP-Complete problem, extending previous hardness results.

Contribution

It establishes NP-Completeness for deciding if the rainbow vertex-connection number is at most any fixed integer k, for all k ≥ 2.

Findings

Deciding if rvc(G) ≤ 2 is NP-Complete.

For every fixed k ≥ 2, the problem is NP-Hard.

The problem belongs to NP for fixed k, making it NP-Complete.

Abstract

A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In a previous paper we showed that it is NP-Complete to decide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that for every integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard. We also show that for any fixed integer $k\geq 2$, this problem belongs to NP-class, and so it becomes NP-Complete.