Fault Tolerant Metric Dimensions of Leafless Cacti Graphs with Application in Supply Chain Management

TL;DR

The paper analyzes fault-tolerant metric dimension $β'(G)$ across bicyclic graphs (types I and II) and extends the results to leafless cactus graphs. It establishes that $β'(G)=4$ for all bicyclic graphs of both types, and derives a concise formula $β'(G)=2(n_1+n_2)$ for leafless cacti, where $n_1$ counts outer cycles and $n_2$ counts even inner cycles with two antipodal common vertices. It also provides a real-world application to supply chain logistics, showing how fault-tolerant landmark sets enable robust node identification even when some landmarks fail. These results offer practical guidance for designing resilient networks with reliable distance-based identification using a compact set of landmarks.

Abstract

A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property. A resolving set is a fault tolerant resolving set if the removal of any vertex from the subset still leaves it a resolving set. The smallest set satisfying this property is the fault tolerant metric basis, and the cardinality of this set is termed as fault tolerant metric dimension of $G$, denoted by $β'(G)$. In this article, we determine the fault tolerant metric dimension of bicyclic graphs of type-I and II and show that it is always $4$ for both types of graphs. We then use these results to form our basis to consider leafless cacti graphs, and calculate their fault tolerant metric dimensions in terms of \textit{inner cycles} and \textit{outer cycles}. We then consider a detailed real world example of supply and distribution center management, and discuss the application of fault tolerant metric dimension in such a scenario. We also briefly discuss some other scenarios where leafless cacti graphs can be used to model real world problems.

Figures (16)

• Figure 3.1: Relationship of common vertex $\mathcal{v}_{\mathcal{n}} =\mathcal{v}_{\mathcal{n}+\mathcal{m}}$ to metric basis of base bicyclic graph $\mathcal{C}_{\mathcal{n},\mathcal{m}}$ of type-I.
• Figure 3.2: Relationship of $\left \{ \mathcal{v}_{\frac{\mathcal{n}}{2}}, \mathcal{v}_{\frac{\mathcal{n}+m}{2}}\right\}$ and metric basis of base bicyclic graph of type-I for even $\mathcal{n},\mathcal{m}$.
• Figure 3.3: $\mathcal{v}_{\mathcal{a}},\mathcal{v}_{\mathcal{b}} \in \mathcal{C}_{\mathcal{n}}$ for base bicyclic graph $\mathcal{C}_{\mathcal{n},\mathcal{m}}$ of type-I for odd $\mathcal{n},\mathcal{m}$
• Figure 3.4: $\mathcal{v}_{\mathcal{a}},\mathcal{v}_{\mathcal{b}} \in \mathcal{C}_{\mathcal{m}}$ for base bicyclic graph $\mathcal{C}_{\mathcal{n},\mathcal{m}}$ of type-I for odd $\mathcal{n},\mathcal{m}$ and $\mathcal{v}_{\mathcal{j}}$ on $\mathcal{P}_{\mathcal{u}\mathcal{v}}$ through $\mathcal{v}_{\mathcal{n}}$
• Figure 3.5: $\mathcal{v}_{\mathcal{a}},\mathcal{v}_{\mathcal{b}} \in \mathcal{C}_{\mathcal{m}}$ for base bicyclic graph $\mathcal{C}_{\mathcal{n},\mathcal{m}}$ of type-I for odd $\mathcal{n},\mathcal{m}$ and $\mathcal{v}_{\mathcal{j}}$ on $\mathcal{v}_{\mathcal{a}} \mathcal{v}_{\mathcal{b}}$-path through $\mathcal{v}_{\mathcal{n}+\lfloor \frac{\mathcal{m}}{2} \rfloor }$.

Theorems & Definitions (49)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof