View PDF HTML (experimental)

Abstract: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.

Submission history

From: Tonny K B [view email]
[v1] Mon, 16 Feb 2026 06:32:41 UTC (15 KB)
[v2] Tue, 3 Mar 2026 10:24:22 UTC (17 KB)
[v3] Tue, 30 Jun 2026 14:07:34 UTC (24 KB)