Local fractional metric dimension of rotationally symmetric planar graphs arisen from planar chorded cycles

TL;DR

The paper introduces the family $\mathcal{G}^m(G)$ of rotationally symmetric planar graphs obtained by edge coalescence of $m$ disjoint planar chorded cycles of order $n$, and analyzes their local fractional metric dimension $ldim_{\mathrm{f}}$ for $n\le 6$. Using linear programming formulations and resolving neighbourhood analysis, the authors provide exact values or upper bounds for the quadrilateral, pentagonal, and hexagonal base cycles, with a corrected wheel-graph case. The results reveal a mix of unbounded and bounded asymptotic behaviours across different base cycles and parity cases, illustrating the rich variety of $ldim_{\mathrm{f}}$ in these rotationally symmetric planar graphs. The work advances understanding of local fractional metric dimension in structured planar graphs and points to future work on higher-order chorded cycles and tighter lower bounds.

Abstract

In this paper, a new family of rotationally symmetric planar graphs is described based on an edge coalescence of planar chorded cycles. Their local fractional metric dimension is established for those ones arisen from chorded cycles of order up to six. Their asymptotic behaviour enables us to ensure the existence of new families of rotationally symmetric planar graphs with either constant or bounded local fractional dimension.

Figures (8)

• Figure 1: Non-isomorphic planar chorded cycles of order $n\leq 6$.
• Figure 2: Edge coalescences between the same pair of planar chorded cycles of order six.
• Figure 3: Planar graph $G$ such that $\mathrm{ldim}_{\mathrm{f}}(G)=\frac{3}{2}\leq \frac{12}{7}$.
• Figure 4: Quadrilateral, pentagonal and hexagonal chorded cycles.
• Figure 5: The graphs $\mathcal{G}^m(Q_1)$ and $\mathcal{G}^m(Q_2)$.

Theorems & Definitions (16)

• Lemma 2.1: Okamoto2010Benish2018Benish2019
• Example 2.1
• Example 3.1
• Lemma 3.1
• proof
• Proposition 3.1
• proof
• Proposition 4.1: Liu2020
• Proposition 4.2
• proof