Mutual-visibility colouring with two colours is NP-complete for every fixed diameter

Abstract

A mutual-visibility set in a graph $G$ is a set $S$ of vertices such that any two vertices of $S$ are connected by a shortest path with no internal vertex in $S$, and the mutual-visibility chromatic number $χ_μ(G)$ is the minimum number of mutual-visibility sets needed to partition $V(G)$. Babu, Di Stefano and Lakshmanan showed that it is NP-complete to decide whether $χ_μ(G)\le 2$ for graphs of diameter four. We show that the problem remains NP-complete for graphs of diameter $d$, for every fixed $d\ge2$; diameter two is the smallest diameter for which the problem is non-trivial. The proof rests on a reformulation of $χ_μ(G)\le 2$ for diameter-two graphs as the $2$-colourability of an associated hypergraph, and on a reduction from $2$-colourability of $3$-uniform hypergraphs without short cycles.