Computing fault-tolerant metric dimension of graphs using their primary subgraphs

TL;DR

This work addresses computing the fault-tolerant metric dimension (FTMD) for graphs formed by point-attaching primary subgraphs. It introduces attaching fault-tolerant resolving sets and the attaching FTMD of subgraphs, establishing a general lower bound $\\mathop{\\mathrm{fdim}}(G) \\\ge \\sum_i \\mathop{\\mathrm{fdim}}^*(G_i)$ and, under Conditions $\\mathscr{C}_1$ and $\\mathscr{C}_2$, proves the exact formula $\\mathop{\\mathrm{fdim}}(G) = \\sum_i \\mathop{\\mathrm{fdim}}^*(G_i)$. The authors extend the results to rooted products, giving explicit expressions in terms of the FTMD of the primary subgraphs, and discuss special cases such as block graphs and vertex-transitive subgraphs. This modular framework clarifies how subgraph structure and attachment points govern fault-tolerant identifiability, with potential impact on robust network design and error-tolerant systems.

Abstract

The metric dimension of a graph is the cardinality of a minimum resolving set, which is the set of vertices such that the distance representations of every vertex with respect to that set are unique. A fault-tolerant metric basis is a resolving set with a minimum cardinality that continues to resolve the graph even after the removal of any one of its vertices. The fault-tolerant metric dimension is the cardinality of such a fault-tolerant metric basis. In this article, we investigate the fault-tolerant metric dimension of graphs formed through the point-attaching process of primary subgraphs. This process involves connecting smaller subgraphs to specific vertices of a base graph, resulting in a more complex structure. By analyzing the distance properties and connectivity patterns, we establish explicit formulae for the fault-tolerant resolving sets of these composite graphs. Furthermore, we extend our results to specific graph products, such as rooted products. For these products, we determine the fault-tolerant metric dimension in terms of the fault-tolerant metric dimension of the primary subgraphs. Our findings demonstrate how the fault-tolerant dimension is influenced by the structural characteristics of the primary subgraphs and the attaching vertices. These results have potential applications in network design, error correction, and distributed systems, where robustness against vertex failures is crucial.

Figures (7)

• Figure 1: Graph $G$ with attaching vertices
• Figure 2: A graph obtained by point attachment of $G_i$, $i\in [4]$, where $G_i \cong C_8$, $i\in [4]$.
• Figure 3: A graph obtained by point attachment of $G_1\cong C_8$, $G_2\cong K_3$ and $G_3\cong P_3$.
• Figure 4: A block graph $G$ constructed by point attaching $G_1\cong K_3$, $G_2\cong K_6$, $G_3\cong K_4$, $G_4\cong K_3$, and $G_5\cong K_5$.
• Figure 5: A graph $G$ constructed by point attaching $G_1\cong C_7$, $G_2\cong K_6$, $G_3\cong P_3$, $G_4\cong C_4$, and $G_5\cong K_3$.

Theorems & Definitions (17)

• Lemma 1
• proof
• Proposition 2
• proof
• Theorem 3
• proof
• Example 4
• Example 5
• Corollary 1
• proof