Fault-Tolerance of Star Graph Based on Subgraph Fault Pattern

Abstract

Traditional fault tolerability is regularly measured by classical vertex or edge connectivity. Menger’s theorem shows that the number of (edge)-disjoint paths is closely related to (edge) connectivity. Clearly, disjoint paths not only provide alternative routings to tolerate faulty vertices but also avoid communication bottlenecks. Furthermore, disjoint paths can speed up the transmission time by distributing data among disjoint paths. In order to assess the fault tolerance of the network objectively, we aim to extend vertex or edge failures to substructure malfunction. In this paper, we show the maximum number of vertex (edge)-disjoint paths in star graph in the case of genetic substructure faults. Let [Formula: see text] ([Formula: see text]) be a [Formula: see text]-dimensional substar of [Formula: see text]. We show that there exist [Formula: see text] vertex (edge)-disjoint paths to connect any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text] is the degree of vertex [Formula: see text] in [Formula: see text]. In addition, we show that (edge) connectivity and [Formula: see text]-extra connectivity of [Formula: see text] are [Formula: see text], [Formula: see text], respectively.