Extensions of networks with given diameter

Abstract This article deals with combinatorial problems motivated by the design of large interconnection networks, in particular how to extend a network by adding nodes while keeping the degree and diameter small. We consider D -admissible extensions in which nodes are added one by one while the diameter remains constant. A D -admissible extension from a graph G to a graph G ' is a sequence of graphs G = G 0 , G 1 ,..., G i ,..., G k = G ', where G i is a subgraph of G i +1 , ¦ V ( G i +1 )¦ = ¦ V ( G i )¦ + 1 and all of the G i have diameter at most D . Furthermore we insist that some of the G i are among the largest of the known graphs with maximum degree and diameter constant. We show that there exist D -admissible extensions from the hypercube of degree d to the hypercube of degree d + 1. Then we study D -admissible extension from the de Bruijn graph UB( d , D ) [resp. Kautz graph UK( d , D )] of maximum degree 2 d and diameter D to UB( d + 1, D ) [resp. UK( d + 1, D )], and show that such D -admissible extensions exist if D = 2, but do not exist if D >2 and d >4.