Extra connectivity of networks modeled by the strong product graphs

Abstract

Given a non-negative integer g, the g-extraconnectivity in a connected graph G is defined as the minimum number of vertices that, when removed, results in G becoming disconnected and each remaining component containing more than g vertices. As an extension of the connectivity, the g-extraconnectivity provides a better measurement for the fault tolerance of an interconnection network. The strong product G1⊠G2 of graphs G1 and G2 is the graph with vertex set V(G1)×V(G2), where two distinct vertices (a1,b1) and (a2,b2) are adjacent if and only if a1=a2 and b1b2∈E(G2), or b1=b2 and a1a2∈E(G1), or a1a2∈E(G1) and b1b2∈E(G2). In this paper, we focus on networks modeled by the strongproductG1⊠G2. We determine the g(≤3)-extraconnectivity of G1⊠G2, where G1 and G2 are regular and maximally connected graphs with girth at least g+4. Additionally, we give the g(≤3)-extra conditional diagnosability of G1⊠G2 under PMC model.