Metric Dimension of Generalized Theta Graphs

TL;DR

The paper investigates the metric dimension of generalized theta graphs, extending classic results for cycles to families $\Theta(s_1, \ldots, s_m)$ with $m$ internally disjoint $P_i$ paths. It introduces the Identical Path Set framework and vector representations to analyze resolving sets, establishing general bounds $m-3 \le \beta(\Theta(s_1, \ldots, s_m)) \le m$ and providing exact results for multiplicities 3 and 4. For $m=3$, the metric dimension is $2$ except in two parity-structured exceptions, yielding $\beta=3$ in those cases; for $m=4$, the authors give a complete classification with four small graphs achieving $\beta=4$, two parameter regimes attaining $\beta=2$, and all remaining cases yielding $\beta=3$. These results elucidate the interplay between path lengths, parity, and path counts in determining metric dimension and offer precise structural insights for generalized theta graphs.

Abstract

A vertex $w$ in a graph $G$ is said to resolve two vertices $u$ and $v$ if $d(w,u)\neq d(w, v)$. A set $W$ of vertices is a resolving set for $G$ if every pair of distinct vertices is resolved by some vertex in $W$. The metric dimension of $G$ is the minimum cardinality of such a set. In this paper, we investigate the metric dimension of generalized theta graphs, providing exact values and structural insights for several subclasses.

Figures (6)

• Figure 1: Generalized Theta Graph
• Figure 2: The vertices on a Generalized Theta Graph within $P_i$
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Proposition 2.2
• Corollary 2.3
• Proposition 2.4
• proof
• Theorem 3.1