Diameter vulnerability of GC graphs

Concern over fault tolerance in the design of interconnection networks has stimulated interest in finding large graphs with maximum degree ∆ and diameter D such that the subgraphs obtained by deleting any set of s vertices have diameter at most D′, this value being close to D or even equal to it. This is the so-called (∆, D, D′, s)-problem. The purpose of this work has been to study this problem for s = 1 on some families of generalized compound graphs. These graphs were designed by one of the authors in [15] as a contribution to the (∆, D)-problem, that is, to the construction of graphs having maximum degree ∆, diameter D and an order large enough. When approaching the mentioned problem in these graphs, we realized that each of them could be redefined as a compound graph, the main graph being the underlying graph of a certain iterated line digraph. In fact, this new characterization has been the key point to prove in a suitable way that the graphs belonging to these families are solutions to the (∆, D, D + 1, 1)-problem. MSC: 05C40, 05C12, 05C20