Unpaired Many-to-Many Disjoint Path Cover of Balanced Hypercubes

2021 ◽  
pp. 1-14
Author(s):  
Huazhong Lü ◽  
Tingzeng Wu
Keyword(s):  
Network Topology ◽  
Upper Bound ◽  
Interconnection Network ◽  
Disjoint Paths ◽  
Disjoint Path ◽  
Path Cover ◽  
Balanced Hypercube ◽  
Vertex Disjoint ◽  
Appl Math ◽  
Source And Sink

A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC) of a graph [Formula: see text] is a set of [Formula: see text] vertex-disjoint paths joining [Formula: see text] distinct pairs of source and sink in which each vertex of [Formula: see text] is contained exactly once in a path. The balanced hypercube [Formula: see text], a variant of the hypercube, was introduced as a desired interconnection network topology. Let [Formula: see text] and [Formula: see text] be any two sets of vertices in different partite sets of [Formula: see text] ([Formula: see text]). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of [Formula: see text] when [Formula: see text]. In this paper, we prove that there exists unpaired many-to-many [Formula: see text]-disjoint path cover of [Formula: see text] ([Formula: see text]) from [Formula: see text] to [Formula: see text], which has improved some known results. The upper bound [Formula: see text] is best possible in terms of the number of disjoint paths in unpaired many-to-many [Formula: see text]-DPC of [Formula: see text].

scholarly journals Torus Pairwise Disjoint-Path Routing

Sensors ◽  
2018 ◽  
Vol 18 (11) ◽  
pp. 3912 ◽  
Author(s):  
Antoine Bossard ◽  
Keiichi Kaneko
Keyword(s):  
Pairwise Disjoint ◽  
Interconnection Network ◽  
Fundamental Problem ◽  
Empirical Evaluation ◽  
Disjoint Paths ◽  
Disjoint Path ◽  
Worst Case ◽  
Routing Problem ◽  
Blue Gene ◽  
Vertex Disjoint

Modern supercomputers include hundreds of thousands of processors and they are thus massively parallel systems. The interconnection network of a system is in charge of mutually connecting these processors. Recently, the torus has become a very popular interconnection network topology. For example, the Fujitsu K, IBM Blue Gene/L, IBM Blue Gene/P, and Cray Titan supercomputers all rely on this topology. The pairwise disjoint-path routing problem in a torus network is addressed in this paper. This fundamental problem consists of the selection of mutually vertex disjoint paths between given vertex pairs. Proposing a solution to this problem has critical implications, such as increased system dependability and more efficient data transfers, and provides concrete implementation of green and sustainable computing as well as security, privacy, and trust, for instance, for the Internet of Things (IoT). Then, the correctness and complexities of the proposed routing algorithm are formally established. Precisely, in an n-dimensional k-ary torus ( n < k , k ≥ 5 ), the proposed algorithm connects c ( c ≤ n ) vertex pairs with mutually vertex-disjoint paths of lengths at most 2 k ( c − 1 ) + n ⌊ k / 2 ⌋ , and the worst-case time complexity of the algorithm is O ( n c 4 ) . Finally, empirical evaluation of the proposed algorithm is conducted in order to inspect its practical behavior.

Unpaired Many-to-Many Disjoint Path Covers on Bipartite k-Ary n-Cube Networks with Faulty Elements

2020 ◽  
Vol 31 (03) ◽  
pp. 371-383
Author(s):  
Jing Li ◽  
Chris Melekian ◽  
Shurong Zuo ◽  
Eddie Cheng
Keyword(s):  
Distributed Systems ◽  
Interconnection Networks ◽  
Disjoint Paths ◽  
Equal Size ◽  
Disjoint Path ◽  
Path Cover ◽  
The Many ◽  
Vertex Disjoint ◽  
Vertex Sets ◽  
Different Parts

The [Formula: see text]-ary [Formula: see text]-cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC for short) of a graph is a set of [Formula: see text] vertex-disjoint paths joining two disjoint vertex sets [Formula: see text] and [Formula: see text] of equal size [Formula: see text] that altogether cover every vertex of the graph. The many-to-many [Formula: see text]-DPC is classified as paired if each source in [Formula: see text] is further required to be paired with a specific sink in [Formula: see text], or unpaired otherwise. In this paper, we consider the unpaired many-to-many [Formula: see text]-DPC problem of faulty bipartite [Formula: see text]-ary [Formula: see text]-cube networks [Formula: see text], where the sets [Formula: see text] and [Formula: see text] are chosen in different parts of the bipartition. We show that, every bipartite [Formula: see text], under the condition that [Formula: see text] or less faulty edges are removed, has an unpaired many-to-many [Formula: see text]-DPC for any [Formula: see text] and [Formula: see text] subject to [Formula: see text]. The bound [Formula: see text] is tight here.

Internally Vertex-Disjoint Paths in Crossed Cube-Connected Ring Networks

2013 ◽  
Vol 321-324 ◽  
pp. 2715-2720
Author(s):  
Xin Yu ◽  
Gao Cai Wang ◽  
Yan Yu
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Disjoint Paths ◽  
Ring Networks ◽  
Vertex Disjoint ◽  
Crossed Cube

Crossed cube is a variation of hypercube, but some properties of the former are superior to those of the latter. However, it is difficult to extend the scale of crossed cube networks. As a kind of hierarchical ring interconnection networks, crossed cube-connected ring interconnection network CRN can effectively overcome the disadvantage. Hence, it is a good topology for interconnection networks. In this paper, we prove that there exist n internally vertex-disjoint paths between any two vertexes in CRN, and analyze the lengths of the paths.

Fault-Tolerant Maximal Local-Connectivity on Cayley Graphs Generated by Transpositions

2019 ◽  
Vol 30 (08) ◽  
pp. 1301-1315 ◽  
Author(s):  
Liqiong Xu ◽  
Shuming Zhou ◽  
Weihua Yang
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Local Connectivity ◽  
Communication Links ◽  
Restricted Condition ◽  
Vertex Disjoint

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A graph [Formula: see text] is said to be maximally local-connected if each pair of vertices [Formula: see text] and [Formula: see text] are connected by [Formula: see text] vertex-disjoint paths. In this paper, we show that Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected and are also [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have a triangle, [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have no triangles. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected if their corresponding transposition generating graphs have no triangles.

scholarly journals Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover

10.37236/170 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
K. B. Wong
Keyword(s):  
Disjoint Paths ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Path Cover ◽  
Minimum Path ◽  
Matching Polynomial ◽  
Minimum Number ◽  
Necessary And Sufficient ◽  
Vertex Disjoint ◽  
Maximum Multiplicity

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.

scholarly journals Evaluating on Topology Survivability Based on Largest Number of Node-Disjoint Paths

2020 ◽  
Vol 2 (3) ◽  
pp. 46
Author(s):  
Qiurong Chen
Keyword(s):  
Network Topology ◽  
Point Of View ◽  
Disjoint Paths ◽  
Disjoint Path ◽  
Node Pair ◽  
Global Topology

<p>On the point of view of Largest Number of Node-Disjoint Path (LNNDP for short) between a node pair in a network, this article states the importance of LNNDP to global survivability of topology at first, then proposes an algorithm to compute maximal number of node-disjoint paths between node pairs. A new topology survivability metric based on LNNDP is put forward to evaluate the global survivability of network topology. It can be used to evaluate the survivability of topology provided. This metric can express accurately global topology survivability.</p>

scholarly journals The m-Path Cover Polynomial of a Graph and a Model for General Coefficient Linear Recurrences

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
John P. McSorley ◽  
Philip Feinsilver
Keyword(s):  
General Term ◽  
Simple Graph ◽  
Time Dependent ◽  
Disjoint Paths ◽  
Third Way ◽  
Linear Recurrences ◽  
Path Cover ◽  
The Matrix ◽  
Cover Polynomial ◽  
Vertex Disjoint

An m-path cover Γ={Pℓ1,Pℓ2,…,Pℓr} of a simple graph G is a set of vertex disjoint paths of G, each with ℓk≤m vertices, that span G. With every Pℓ we associate a weight, ω(Pℓ), and define the weight of Γ to be ω(Γ)=∏k=1r‍ω(Pℓk). The m-path cover polynomial of G is then defined as ℙm(G)=∑Γ‍ω(Γ), where the sum is taken over all m-path covers Γ of G. This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the m-path cover polynomial of a weighted path P(m-1,n) and find the (m+1)-term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the m-path cover polynomial of a suitably weighted cycle C(n). A directed graph, T(m), the edge-weighted m-trellis, is introduced and so a third way to generate the solutions to the above (m+1)-term recurrence is presented. We also give a model for general-term linear recurrences and time-dependent Markov chains.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

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