  The concept of mutual visibility in graphs, introduced recently, addresses a
fundamental problem in Graph Theory concerning the identification of the
largest set of vertices in a graph such that any two vertices have a shortest
path connecting them, excluding internal vertices of the set. Originally
motivated by some challenges in Computer Science related to robot navigation,
the problem seeks to ensure unobstructed communication channels between
navigating entities. The mutual-visibility problem involves determining a
largest mutual-visibility set in a graph. The mutual-visibility number of a
graph represents the cardinality of the largest mutual-visibility set. This
concept has sparked significant research interest, leading to connections with
classical combinatorial problems like the Zarankiewicz problem and Tur\'an-type
problems. In this paper, we consider practical limitations in network
visibility and our investigation extends the original concept to $k$-distance
mutual-visibility. In this case, a pair of vertices is considered $S$-visible
if a shortest path of length at most $k$ exists, excluding internal vertices
belonging to the set $S$. The $k$-distance mutual-visibility number represents
the cardinality of a largest $k$-distance mutual-visibility set. We initiate
the study of this new graph parameter. We prove that the associate decision
problem belongs to the NP-complete class. We also give some properties and
tight bounds, as well as, the exact value of such parameter for some particular
non trivial graph classes.
