The Equidistant Dimension of Corona Product Graphs

Abstract

A subset $S$ of vertices, in a connected graph $G$, is called a distance-equalizer set if for every pair of distinct vertices outside $S$, there exists a vertex in $S$ equidistant to both. The equidistant dimension, denoted by $ξ(G)$, is defined as the minimum cardinality of such sets. While several distance-based parameters have been studied for different graph products, the equidistant dimension of corona product graphs has remained unexplored. In this paper, we investigate the equidistant dimension of the corona product $G \odot H$ of two graphs $G$ and $H$. We introduce the empty bisector graph $\widehat{G}$, an auxiliary construction that relates pairs of vertices in $G$ that cannot be equidistant from any third vertex. Using this framework, we establish tight bounds on the equidistant dimension of $G \odot H$ and derive exact values for several classical families of graphs. Moreover, we show that for any fixed base graph $G$, the equidistant dimension of $G \odot H$ depends on $H$ only through its order and eventually becomes linear in $\n(H)$.

Figures (7)

• Figure 1: Graph $G$ used to illustrate forward-equalized pairs.
• Figure 2: Corona product $C_3\odot P_2$, for which there exists a $\xi(C_3\odot P_2)$-set $S=\{2,3,4\}$ that does not satisfy Lemma \ref{['structHindep']}.
• Figure 3: A non-bipartite graph $G$ (to the left) with $\beta^*(G) > 0$, and its corresponding empty bisector graph $\widehat{G}$ (to the right). In this example the lower bound stated in Theorem \ref{['improvedlbound']} is attained.
• Figure 4: A Graph $G$ (to the left), and its corresponding empty bisector graph $\widehat{G}$ (to the right), such that $\xi(G \odot H) = \beta(\widehat{G})\mathop{\mathrm{n}}\nolimits(H)+ \mathop{\mathrm{n}}\nolimits(G)$ for every graph $H$ with $\mathop{\mathrm{n}}\nolimits(H) \ge 2$, but $\xi(G \odot H) < \beta(\widehat{G})\mathop{\mathrm{n}}\nolimits(H) + \mathop{\mathrm{n}}\nolimits(G)$ whenever $\mathop{\mathrm{n}}\nolimits(H) = 1$.
• Figure 5: Graph $G$ (to the left), and its corresponding empty bisector graph $\widehat{G}$ (to the right), with $\beta(\widehat{G}) = 3$, $\alpha(\widehat{G})=5$ and $\beta^*(G) = 0$.

Theorems & Definitions (58)

• Definition 2.1: Distance-equalizer set Gonzalez2022
• Definition 2.2: Total distance-equalizer set GispertFernandez2024
• Definition 2.3: Bisector of two vertices
• Definition 2.4
• Definition 2.5: Cartesian product of graphs Imrich2000
• Definition 2.6: Corona product of graphsHarary1969
• Theorem 2.7: Gallai's Theorem Gallai1959
• Lemma 3.1
• proof
• Lemma 3.2