Mutual-visibility in distance-hereditary graphs: a linear-time algorithm

TL;DR

This paper introduces a linear-time algorithm for computing the mutual-visibility number in distance-hereditary graphs, a problem previously known to be NP-complete for general graphs.

Contribution

It provides the first efficient algorithm for mutual-visibility number in distance-hereditary graphs, expanding the understanding of visibility properties in this class.

Findings

Mutual-visibility number can be computed in linear time for distance-hereditary graphs.

The problem is NP-complete for general graphs but tractable for this class.

The paper extends known formulas to a broader class of graphs.

Abstract

The concept of mutual-visibility in graphs has been recently introduced. If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u, v\}$. If every two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$. It is known that computing the mutual-visibility number of a graph is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and grids. In this paper, we study the mutual-visibility in distance-hereditary graphs and show that the mutual-visibility number can be computed in linear time for this class.