Cartesian Products Of Trees Research Articles

Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

Zip product of graphs and crossing numbers

AbstractD. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a weaker condition. Based on the new condition, we then establish some general expressions for bounding the crossing numbers of (capped) Cartesian product of trees with graphs (possibly without dominating vertex). Exact values of the crossing numbers of Cartesian product of trees with most graphs of order at most five are obtained by applying these expressions, which extend some previous results due to M. Klešč. In fact, our results can also be applied to deal with Cartesian product of trees with graphs of order more than five.

Embedding hypercubes and folded hypercubes onto Cartesian product of certain trees

The hypercube network is one of the most popular interconnection networks since it has simple structure and is easy to implement. The folded hypercube is an important variation of the hypercube. Interconnection networks play a major role in the performance of distributed memory multiprocessors and the one primary concern for choosing an appropriate interconnection network is the graph embedding ability. A graph embedding of a guest graph G into a host graph H is an injective map on the vertices such that each edge of G is mapped into a path of H. The wirelength of this embedding is defined to be the sum of the lengths of the paths corresponding to the edges of G. In this paper we embed hypercube and folded hypercube onto Cartesian product of trees such as 1-rooted complete binary tree and path, sibling tree and path to minimize the wirelength.

Cartesian products of trees and paths

We characterize the (weak) Cartesian products of trees among median graphs by a forbidden 5-vertex convex subgraph. The number of tree factors (if finite) is half the length of a largest isometric cycle. Then a characterization of Cartesian products of n trees obtains in terms of isometric cycles and intervals. Finally we investigate to what extent the proper intervals determine the product structure. © 1996 John Wiley & Sons, Inc.

Cartesian products of trees and paths

Cartesian products of trees and paths

Cartesian products of block graphs

Hamming graphs (being the Cartesian products of complete graphs) are known to be the quasi-median graphs not containing the 3-vertex path as a convex subgraph. Similarly, the Cartesian products of trees have been characterized among median graphs by a single forbidden convex subgraph. In this note a common generalization of these two results is given that characterizes the Cartesian products of block graphs.

Harmonic functions on cartesian products of trees with finite graphs

Let G be a graph which is the Cartesian product of an infinite, locally finite tree T and a finite, connected graph A . On G , consider a stochastic transition operator P giving rise to a transient random walk and such that positive transitions occur only along the edges of G . We construct a matrix-valued kernel on T , which extends naturally in the second variable to the space of ends Ω of T . This kernel is used to derive a unique integral representation over Ω of all—not necessarily positive—functions on G which are harmonic with respect to P . We explain the relation with the Martin boundary and the positive harmonic functions and, as a particular case, we show what happens when A arises from a finite abelian group and P is compatible with the structure of A .

On a game of policemen and robber

We consider a game where policemen try to catch a robber on a graph G (as defined by A. Quilliot) and we find the exact minimal number of policemen needed when G is a Cartesian product of trees.