Wide-diameter of Product Graphs

Abstract

The product graph B * F of graphs B and F is an interesting model in the design of large reliable networks. Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width k of a graph G, k-diameter, is defined as the minimum integer d for which there exist at least k internally disjoint paths of length at most d between any two distinct vertices in G. Denote by $\cal{D}^W_c (G)$ the c-diameter of G and κ(G) the connectivity of G. We prove that $\cal{D}^W_{a+b}(B * F) \le r_a(F) + \cal{D}^W_b (B) + 1$ for a ≤ κ(F) and b ≤ κ(B). The Rabin number r_c(G) is the minimum integer d such that there are c internally disjoint paths of length at most d from any vertex v to any set of c vertices {v₁, v₂, ... , v_c}.