The vertex visibility number of graphs

2025-10-22

AI Summary

This paper introduces and studies the *vertex visibility number* of a graph: for a vertex $x$ in a graph $G$, $v_x(G)$ is defined as the maximum number of leaves in a shortest-path tree rooted at $x$; the vertex visibility number of $G$ is $vv(G) = max_x v_x(G)$. The problem addresses the computational complexity, structural characterization, and exact evaluation of this novel graph invariant. Methodologically, the work combines shortest-path analysis, combinatorial optimization, and structural graph theory. Key contributions include: (i) formal definition and rigorous characterization of the *vertex visibility set*, establishing its equivalence to the leaf set of a shortest-path tree; (ii) proof that deciding whether $v_x(G) geq k$ is NP-complete; (iii) derivation of general upper and lower bounds for $vv(G)$; and (iv) exact computation of $vv(G)$ for Cartesian products, grid graphs, square prisms, and square tori. The results establish vertex visibility number as a new structural metric quantifying graph “visual accessibility” from individual vertices.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Security and Privacy: Large-scale security measurements

Abstract

If $xin V(G)$, then $Ssubseteq V(G)setminus{x}$ is an $x$-visibility set if for any $yin S$ 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 ${ m vv}(G)$ of $G$. It is proved that ${ m vv}(G)$ 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 $xin V(G)$, 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.

Problem

• Determining maximum vertex visibility sets in graphs
• Proving NP-completeness of visibility set decision problem
• Establishing bounds for vertex visibility in Cartesian products

Innovation

• Defines vertex visibility number via shortest paths
• Proves NP-completeness for diameter two graphs
• Determines exact values for Cartesian product graphs