Characterizing and computing in linear time mutual-visibility parameters in distance-hereditary graphs

Abstract

The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $X\subseteq V\left(G\right)$ so that for any two vertices $x,y\in X$ there is a shortest $x,y$-path whose internal vertices are all not in $X$. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $X$. It is known that solving the mutual-visibility problem in all its variations is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and for the Cartesian product of some simple graphs like paths, cliques and cycles.

In this paper, we study the (variations of) mutual-visibility problem in the context of distance-hereditary graphs. In particular, we introduce the direct canonical decomposition of a graph as a tool for defining useful structural properties of the graphs studied. Then, we show that such properties allow us to devise a linear-time algorithm for solving all the variants of the mutual-visibility problem for distance-hereditary graphs. In turn, this allowed us to show that a recently posed conjecture about the total mutual-visibility number of distance-hereditary graphs holds.