The vertex visibility number of graphs

TL;DR

This paper introduces the vertex visibility number of graphs, explores its properties, computational complexity, bounds, and exact values for specific graph classes, advancing understanding of graph visibility parameters.

Contribution

It defines the vertex visibility number, proves its relation to shortest-path trees, establishes NP-completeness, and determines bounds and exact values for certain graph families.

Findings

Vertex visibility number equals the maximum number of leaves in a shortest-path tree.

Deciding if v_x(G) ≥ k is NP-complete for diameter-2 graphs.

Exact values are found for square grids, prisms, and toruses.

Abstract

If $x\in V(G)$, then $S\subseteq V(G)\setminus\{x\}$ is an $x$-visibility set if for any $y\in S$ there exists a shortest $x,y$-path avoiding $S$. The $x$-visibility number $v_x(G)$ is the maximum cardinality of an $x$-visibility set, and the maximum value of $v_x(G)$ among all vertices $x$ of $G$ is the vertex visibility number ${\rm vv}(G)$ of $G$. It is proved that ${\rm vv}(G)$ is equal to the largest possible number of leaves of a shortest-path tree of $G$. Deciding whether $v_x(G) \ge k$ holds for given $G$, a vertex $x\in V(G)$, and a positive integer $k$ is NP-complete even for graphs of diameter $2$. Several general sharp lower and upper bounds on the vertex visibility number are proved. The vertex visibility number of Cartesian products is also bounded from below and above, and the exact value of the vertex visibility number is determined for square grids, square prisms, and square toruses.