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 Δ 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 (Δ,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 Gomez (Ars Combin. 29-B (1990) 33) as a contribution to the (Δ,D)-problem, that is, to the construction of graphs having maximum degree Δ, 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 (Δ,D,D + 1, 1)-problem.

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