Constructing vertex–disjoint paths in (n, k)-star graphs

This work describes a novel routing algorithm for constructing a container of width n − 1 between a pair of vertices in an (n, k)-star graph with connectivity n − 1. Since Lin et al. [T.C. Lin, D.R. Duh, H.C. Cheng, Wide diameter of (n, k)-star networks, in: Proceedings of the International Conference on Computing, Communications and Control Technologies, vol. 5, 2004, pp. 160–165] already calculated the wide diameters in (n, n − 1)-star and (n, 1)-star graphs, this study only considers an (n, k)-star with 2 ⩽ k ⩽ n − 2. The length of the longest container among all constructed containers serves as the upper bound of the wide diameter of an (n, k)-star graph. The lower bound of the wide diameter of an (n, k)-star graph with 2 ⩽ k ⩽ ⌊n/2⌋ and the lower bound of the wide diameter of a regular graph with a connectivity of 2 or above are also computed. Measurement results indicate that the wide diameter of an (n, k)-star graph is its diameter plus 2 for 2 ⩽ k ⩽ ⌊n/2⌋, or its diameter plus a value between 1 and 2 for ⌊n/2⌋ + 1 ⩽ k ⩽ n − 2.