Rija Erveš

Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Mixed fault diameter of a graph G , D ( a , b ) ( G ) , is the maximal diameter of G after deletion of any a vertices and any b edges. Special cases are the (vertex) fault diameter D a V = D ( a , 0 ) and the edge fault diameter D a E = D ( 0 , a ) . Let G be a Cartesian graph bundle with fibre F over the base graph B . We show that(1) D a + b + 1 V ( G ) ? D a V ( F ) + D b V ( B ) when the graphs F and B are k F -connected and k B -connected, 0 < a < k F , 0 < b < k B , and provided that D ( a - 1 , 1 ) ( F ) ? D a V ( F ) and D ( b - 1 , 1 ) ( B ) ? D b V ( B ) and(2) D a + b + 1 E ( G ) ? D a E ( F ) + D b E ( B ) when the graphs F and B are k F -edge connected and k B -edge connected, 0 ? a < k F , 0 ? b < k B , and provided that D a E ( F ) ? 2 and D b E ( B ) ? 2 .

Wide-diameter of Product Graphs

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

The edge fault-diameter of Cartesian graph bundles

\cal{D}^W_c G

Edge, vertex and mixed fault diameters

the c-diameter of G and κG the connectivity of G. We prove that

Mixed fault diameter of Cartesian graph bundles

\cal{D}^W_{a+b}B * F \le r_aF + \cal{D}^W_b B + 1

Mixed connectivity of Cartesian graph products and bundles

for a ≤ κF and b ≤ κB. The Rabin number rcG 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,..., vc}.