Extra Connectivity and Structure Connectivity of 2-Dimensional Torus Networks

Abstract

The 2-dimensional torus networks, denoted by [Formula: see text], is a special case of the famous [Formula: see text]-dimensional torus network. The extra connectivity, structure connectivity and sub-structure connectivity are generalized connectivity to measure the fault-tolerance of the networks. In this paper, we show the 1-extra connectivity of [Formula: see text] for [Formula: see text] and [Formula: see text] and 2-extra connectivity of [Formula: see text] for [Formula: see text]. We also determine the [Formula: see text]-structure connectivity and [Formula: see text]-substructure connectivity of 2-dimensional torus network [Formula: see text] ([Formula: see text] and [Formula: see text]) when the structure subgraph [Formula: see text], where [Formula: see text] and [Formula: see text]. These results are sharp since the connectivity of [Formula: see text] ([Formula: see text]) is 4.