Diameter variability of cycles and tori

Abstract

The diameter of a graph is an important factor for communication as it determines the maximum communication delay between any pair of processors in a network. Graham and Harary [N. Graham, F. Harary, Changing and unchanging the diameter of a hypercube, Discrete Applied Mathematics 37/38 (1992) 265–274] studied how the diameter of hypercubes can be affected by increasing and decreasing edges. They concerned whether the diameter is changed or remains unchanged when the edges are increased or decreased. In this paper, we modify three measures proposed in Graham and Harary (1992) to include the extent of the change of the diameter. Let D - k ( G ) is the least number of edges whose addition to G decreases the diameter by (at least) k, D + 0 ( G ) is the maximum number of edges whose deletion from G does not change the diameter, and D + k ( G ) is the least number of edges whose deletion from G increases the diameter by (at least) k. In this paper, we find the values of D - k ( C m ) , D - 1 ( T m , n ) , D - 2 ( T m , n ) , D + 1 ( T m , n ) , and a lower bound for D + 0 ( T m , n ) where C m be a cycle with m vertices, T m , n be a torus of size m by n.