The vertex visibility number of graphs

TL;DR

This work introduces and rigorously analyzes the vertex visibility number vv(G) via the x-visibility framework. It proves the equivalence between vv(G) and the maximum number of leaves in a shortest-path tree, and establishes NP-completeness of deciding v_x(G) ≥ k even for diameter-2 graphs, tying visibility to fundamental shortest-path structures. The authors derive general lower and upper bounds, identify exact vv-values for several Cartesian product families, and provide explicit results for square grids, prisms, and toruses. They also connect vv to classical graph parameters (e.g., mutual-visibility, MD_G(x), and stress vertices) and discuss implications for future product graphs research. The findings offer precise, computationally relevant insights into vertex visibility and its behavior under graph products, with potential applications in network design and metric graph theory.

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.

Figures (7)

• Figure 1: A graph $G$ with $\ell(G) = 11$ but $\mathop{\mathrm{vv}}\nolimits(G) = \mu_x(G) = 10$ (also attained at $y$ and $z$).
• Figure 2: The graph $G'$ for $G=P_5$.
• Figure 3: vv-sets of $P_6 \,\square\, P_6$ and $P_7 \,\square\, P_7$.
• Figure 4: vv-sets of $P_5 \,\square\, C_5$ and $P_7 \,\square\, C_7$.
• Figure 5: vv-sets of $P_6 \,\square\, C_6$ and $P_8 \,\square\, C_8$.

Theorems & Definitions (25)

• Lemma 1.1
• proof
• Definition 2.1
• Theorem 2.2
• proof
• Proposition 2.3
• Theorem 2.4
• proof
• Proposition 3.1
• Lemma 3.2