On large (Δ, D, D, 1)-graphs

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 ≤ ∆ − 1, where ∆ is the maximum degree of the graph. Graphs of maximum degree ∆, diameter ≤ D and such that the graphs obtained by deletion of up to s vertices have diameter ≤ D′ are known as (∆, D, D′, s) graphs. This paper 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 paper also contains an updated table of the largest known (∆, D, D, 1) graphs, in which most of the entries correspond to the constructions put forward in this paper.