The Sufficient Conditions of (δ(G) − 2)-(|F|-)Fault-Tolerant Maximal Local-(Edge-)Connectivity of Connected Graphs

Abstract

An interconnection network is usually modeled by a connected graph in which vertices represent processors and edges represent links between processors. The connectivity is an important parameter to evaluate the fault tolerance of interconnection networks. A connected graph [Formula: see text] is maximally local-(edge-)connected if each pair vertices [Formula: see text] of [Formula: see text] is connected by min[Formula: see text] pairwise (edge-)disjoint paths between [Formula: see text] and [Formula: see text] in [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant maximally local-(edge-)connected if [Formula: see text] is maximally local-(edge-)connected for any [Formula: see text] ([Formula: see text]) with [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant maximally local-(edge-)connected of order [Formula: see text] if [Formula: see text] is maximally local-(edge-)connected for any [Formula: see text] with [Formula: see text], where [Formula: see text] is a conditional faulty vertex (edge) set of order [Formula: see text]. In this paper, we obtain the sufficient condition of connected graphs to be [Formula: see text]-edge-fault-tolerant maximally local-edge-connected. Moreover, we consider the sufficient condition of connected graphs to be [Formula: see text]-fault-tolerant maximally local-(edge-)connected of order [Formula: see text]. Some previous results in [Theor. Comput. Sci. 731 (2018) 50–67] and [Theor. Comput. Sci. 847 (2020) 39–48] are extended.