The Diameter Variability of the Cartesian Product of Graphs

Abstract

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

Pointed order polytopes: Studying geometrical aspects of the polytope of bi-capacities

“…i) By Remark 4 we know that O(P, a) = O(P 1 , a) × O [−1,1] (P 2 ). Since the adjacency graph of the product of polytopes is the cartesian product of its adjacency graphs and the diameter of the cartesian product of graphs is the sum of their diameters (see [6]), the result holds.…”

Some diameter notions in lexicographic product of graphs

“…where p, q ∈ {1, 2, ...n 2 } and i ∈ {1, 2, ...n 1 } are deleted, the diam(G ′ ) remains the same. Now, let the n 2 2…”

“…D −k (G) is the minimum number of edges to be added to G to decrease the diameter by (at least) k and D 0 (G) is the maximum number of edges that can be deleted from G so that the diameter is not altered. In [1], [2], the diameter variability of the product graphs are discussed. In [12], Wang et al studied the diameter variability of cycles and tori.…”