Removing edges from hypercubes to obtain vertex‐symmetric networks with small diameter

Abstract

The binary hypercube Q _n has a small diameter, but a relatively large number of links. Because of this, efforts have been made to determine the maximum number of links that can be deleted without increasing the diameter. However, the resulting networks are not vertex‐symmetric. We propose a family of vertex‐symmetric spanning subnetworks of Q _n , whose diameter differs from that of Q _n by only a small constant factor. When n=2^k, the cube‐connected cycles network of dimension n is a vertex‐symmetric spanning subnetwork of Q _{n+lg n} . By selectively adding hypercube links, we obtain a degree 6 vertex‐symmetric network with diameter 3n/2. We also introduce a vertex‐symmetric spanning subnetwork of Q _n−1 with degree log₂ n, diameter 3n/2−2, log₂ n‐connectivity and maximal fault tolerance. This network hosts Q _n−1 with dilation 2(log₂ n)−1.