Wide Diameters of Cartesian Product Graphs and Digraphs

Abstract

Abstract.In graph theory and study of fault tolerance and transmission delay of networks, connectivity and diameter of a graph are two very important parameters and have been deeply studied by many authors. Wide diameter combining connectivity with diameter is a more important parameter to measure fault tolerance and efficiency of parallel processing computer networks and has received much attention in the recent years. Diameter with width k of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least k internally disjoint paths of length at most d. In the present paper, the tight upper bounds of wide diameter of the Cartesian product graphs are obtained. Some known results can be deduced or improved from ours.

The vulnerability of the diameter of the enhanced hypercubes

“…These parameters were studied by several authors for some Cartesian product graphs [7,25,26] and for the hypercube and its variants [3,5,8,10,18,19,20].…”

“…They combine connectivity with diameter to measure simultaneously the fault-tolerance and efficiency of parallel processing computer networks. These parameters were studied by several authors for some Cartesian product graphs [7,25,26] and for the hypercube and its variants [3,5,8,10,18,19,20].…”

Property Pd,m and efficient design of reliable networks

“…In many recent papers involving the wide diameter, the focus has been on finding or bounding the wide diameter of particular classes of graphs. See (13,21,22) for examples. Several other measures of recent interest such as the Rabin number, the fault tolerant diameter, and the edge wide diameter also share a strong relation to the concept of property P d , m as they bring together the ideas of diameter and containers between vertices in an attempt to measure the reliability of networks.…”

Construction of vertex-disjoint paths in alternating group networks

“…Duh, Chen, and Hsu [8] investigated combinatorial properties of generalized hypercubes, including best containers, wide diameter, and fault diameter. Xu [26] obtained results on the wide diameters of Cartesian product graphs, a class of interconnection networks that includes the hypercune as a special case.…”