Abstract
Fault tolerance is the characteristic of a system that permits it to carry on its intended operations in case of the failure of one of its units. Such a system is known as the fault-tolerant self-stable system. In graph theory, if we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this paper, we determine the fault-tolerant resolvability in line graphs. As a main result, we computed the fault-tolerant metric dimension of line graphs of necklace and prism graphs (2010 Mathematics Subject Classification: 05C78).
Highlights
Introduction andPreliminaries e concept of metric dimension of the general metric space was presented in 1953
Let G be a graph with fault-tolerant metric dimension 3, and let v1, v2, v3 ⊂ V(G) be a fault-tolerant resolving set in G. en, the degree of each vertex v1, v2, and v3 is at most 3. e rest of the paper is structured as follows: in Section 2, we will compute the fault-tolerant metric dimension of the line graph of the necklace graph
We have studied for the first time the faulttolerant metric dimension of the line graph of a graph
Summary
Introduction andPreliminaries e concept of metric dimension of the general metric space was presented in 1953 (see [1]). Hernando et al in [7] computed the fault-tolerant resolving set for tree graphs and proved that β′(Pn) 2 for the path graph Pn on n ≥ 2 vertices. We determine the fault-tolerant metric dimension in line graphs.
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