Completely independent spanning trees in torus networks

Abstract

Let T 1 , T 2 , . . . , T k be spanning trees in a graph G. If for any two vertices u, v in G, the paths from u to v in T 1 , T 2 , . . . , T k are pairwise internally disjoint, then T 1 , T 2 , . . . , T k are completely independent spanning trees in G. Completely independent spanning trees can be applied to fault-tolerant communication problems in interconnection networks. In this article, we show that there are two completely independent spanning trees in any torus network. Besides, we generalize the result for the Cartesian product. In particular, we show that there are two completely independent spanning trees in the Cartesian product of any 2-connected graphs.

Completely independent spanning trees in some regular graphs

“…Suppose m is odd. For m = 3, Hasunuma and Morisaka [7] has proven that in any Cartesian product of 2-connected graphs, there are two completely independent spanning trees. By Propositions 3.12, 3.13, 3.10, 3.11 and 3.9, we obtain that there exist ⌈m/2⌉ completely independent spanning trees for m = 5 or (m = 7 ∧ n = 3, 4) or (m = 9 ∧ n = 4, 5).…”

Completely independent spanning trees for enhancing the robustness in ad-hoc Networks

“…The decision problem of determining whether there are two Completely Independent Spanning Trees in a graph G is NP-hard in general [20]. CISTs have been studied on different classes of graphs, such as underlying graphs of line graphs [5], maximal planar graphs [20], cartesian product of two cycles [21] and complete graphs, complete bipartite and tripartite graphs [22]. Inspired by the Tutte-Nash-William theorem on the edge-disjoint trees [11], [12], Hasunama conjectured in [20] that there are k completely independent spanning trees in any 2k-connected graph.…”

Completely Independent Spanning Trees on Some Interconnection Networks

“…Hasunuma [8] conjectured that there are k CISTs in any 2k-connected graph and showed the NP-completeness for determining the existence of two CISTs in an arbitrary graph G. So far the study related to CISTs has received less attention except for [1], [8]- [10]. Araki [1] showed that a graph G of n vertices has two CISTs if the minimum degree of G is at least n/2, and the square of a 2-connected graph has two CISTs.…”

“…Araki [1] showed that a graph G of n vertices has two CISTs if the minimum degree of G is at least n/2, and the square of a 2-connected graph has two CISTs. Hasunuma showed that there are k CISTs in the underlying graph of any k-connected line digraph L(G) [8], there are two CISTs in any 4-connected maximal planar graph [9], and there are two CISTs in the Cartesian product of any 2-connected graphs [10]. Recently, Péterfalvi [15] gave counterexamples to disprove Hasunuma's conjecture.…”