Embedding of Cycles and Grids in Star Graphs

Abstract

The use of the star graph as a viable interconnection scheme for parallel computers has been examined by a number of authors in recent times. An attractive feature of this class of graphs is that it has sublogarithmic diameter and has a great deal of symmetry akin to the binary hypercube. In this paper we describe a new class of algorithms for embedding (a) Hamiltonian cycle (b) the set of all even cycles and (c) a variety of two- and multi-dimensional grids in a star graph. In addition, we also derive an algorithm for the ranking and the unranking problem with respect to the Hamiltonian cycle.

Hamiltonian-laceability of star graphs

“…They possess many nice topological properties, for example, recursiveness, vertex and edge symmetry, maximal fault tolerance, sublogarithmic degree and diameter, and strong resilience [1,2], which are all desirable in building interconnection topologies for parallel and distributed systems. The star graphs were shown able to embed cycles [13,19,26], grids [13,19,21], trees [4,7,12], and hypercubes [17]. Their fault diameters were computed in [14,22,23].…”

“…In [13], Jwo et al showed that the star graphs are bipartite. Besides, the two partite sets have equal size.…”

Performance Modelling of Necklace Hypercubes

“…A large number of interconnection networks have been proposed and studied for highly parallel distributedmemory multicomputers [3,7,8,11,14,15,17,18,26,29,30,31]. Among them the hypercube has been one of the most famous ones which has many desirable properties such as logarithmic diameter and fault-tolerance.…”

Embedding longest fault-free paths in arrangement graphs with faulty vertices

“…Embedding cycles and paths in the star graph with or without faults has been extensively studied in the literature [12,14,[15][16][17][18][19]23,27]. On the other hand, it was shown in [10] that when n − k ≥ 2 the arrangement graph is pancyclic, that is, contains cycles of all possible lengths.…”