Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r(v|Π)=(d(v,Si))i=1k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P(G) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

中文翻译：

---

关于图的容错分区维

容错解析分区是解析分区的自然扩展，解析分区在计算机科学的不同领域具有许多应用，例如传感器网络，智能系统，优化和机器人导航。对于非平凡的连通图G（V（G），E（G）），顶点v相对于V（G）的有序分区Si = {Si：1≤i≤k}的分区表示为k-向量r（v |Π）=（d（v，Si））i = 1k，其中，d（v，Si）= min {d（v，x）| x∈Si}，对于i∈{1，2 ，…，k}。如果对于所有u≠v∈V（G），r（u |Π）和r（v |Π）至少相差两个位置，则分区被称为G的容错分区解析集。最小基数的容错分区解析集称为G的容错分区基础，其基数称为G的容错分区维，用P（G）表示。在这篇文章中，