Concern about fault tolerance in the design of interconnection networks has aroused interest in finding large graphs such that the subgraphs obtained by deleting any set of up to s vertices have small diameter. Clearly, 1 ≤ s ≤ Δ − 1, where Δ is the maximum degree of the graph. Graphs of maximum degree Δ, diameter ≤ D and such that the graphs obtained by deletion of up to s vertices have diameter ≤ D′ are known as (Δ, D, D′, s)-graphs. This article considers the case s = 1 and D = D′. In other words, it deals with the search for large graphs whose diameter does not increase after deleting one vertex. The article also contains an updated table of the largest known (Δ, D, D, 1)-graphs, in which most of the entries correspond to the constructions put forward in this article. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 316–327 2011
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