k-Mutual Visibility in Graphs

In this paper, we introduce the notion of $k$-mutual visibility, a relaxation of classical mutual visibility in which every pair of selected vertices is joined by a shortest path containing at most $k$ internal vertices of the selected set.

This parameterized concept naturally generalizes classical mutual visibility and provides a graded notion of obstruction tolerance.

We define the $k$-mutual visibility number $\mu_k(G)$ and establish its fundamental properties.

We derive bounds on $\mu_k(G)$ in terms of graph parameters such as diameter and girth, and determine its exact value for several fundamental graph classes.

We further investigate $k$-mutual visibility in convex subgraphs and characterize it in block graphs by introducing the notion of $k$-admissible sets in the associated block-cutpoint tree.

We present a polynomial-time algorithm, kMV, that recognizes whether a given subset $S\subseteq V(G)$ is a $k$-mutual visibility set of $G$.

We also formulate the $k$-Mutual Visibility decision problem and prove that it is NP-complete.

Finally, we define the $k$-mutual visibility covering number $\tau_k(G)$ and establish several of its fundamental properties.