Abstract

The ability to uniquely identify all nodes in a network based on network distances has proven to be highly beneficial despite the computational challenges involved in discovering minimal resolving sets within an interconnection network. A subset R of vertices of a graph G is referred to as a resolving set of the graph if each node can be uniquely identified by its distance code with respect to R, with its minimal cardinality defining the metric dimension of G. Similarly, a resolving set \(F \subseteq V\) is designated as a fault-tolerant resolving set if \(F {\setminus } \{s\}\) serves as a resolving set for each \(s \in F\) . The minimum cardinality of F represents the fault-tolerant metric dimension of G. Although determining the precise metric dimension of a given graph remains challenging, various effective techniques and meaningful constraints have been developed for different graph families. However, no notable technique has been developed to find fault-tolerant metric dimension of a given graph. Recently, Prabhu et al. have shown that each twin vertex of G belongs to every fault-tolerant resolving set of G. Consequently, the fault-tolerant metric dimension is equal to the order of the graph G if and only if each vertex of G is a twin vertex, a characterization proved in Appl Math Comput 420:126897, 2022, corrects a wrong characterization in the literature. It is also interesting to note from the above literature correction that the twin vertices are necessary to form the fault-tolerant resolving set, but determining whether they are sufficient is challenging. Evidence of this context is also discussed in this paper through the amalgamation of perfect binary trees. This article focuses on determining the exact value of the fault-tolerant metric dimension of generalized fat trees. For the amalgamation of perfect binary trees, both the metric dimension and fault-tolerant metric dimension were precisely found.

中文翻译：

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广义胖树的容错基础和完美二叉树派生架构

摘要

尽管在互连网络中发现最小解析集存在计算挑战，但基于网络距离唯一识别网络中所有节点的能力已被证明是非常有益的。如果每个节点可以通过其相对于R的距离代码唯一标识，并且其最小基数定义G的度量维度，则图G的顶点子集R被称为图的解析集。类似地，如果\(F {\setminus } \{s\}\)作为F 中每个 \(s \)的解析集，则解析集\(F \subseteq V\)被指定为容错解析集\)。F的最小基数表示G的容错度量维度。尽管确定给定图的精确度量维度仍然具有挑战性，但已经针对不同的图族开发了各种有效的技术和有意义的约束。然而，还没有开发出显着的技术来查找给定图的容错度量维度。最近，普拉布等人。已经证明G的每个孪生顶点都属于G的每个容错解析集。因此，容错度量维度等于图G的阶当且仅当G的每个顶点都是双顶点时，Appl Math Comput 420:126897, 2022 中证明的表征纠正了文献中的错误表征。有趣的是，从上述文献修正中注意到，孪生顶点对于形成容错解析集是必要的，但确定它们是否足够是具有挑战性的。本文还通过完美二叉树的合并讨论了这种背景的证据。本文重点讨论确定广义胖树容错度量维度的准确值。对于完美二叉树的合并，精确地找到了度量维数和容错度量维数。