Gossiping in vertex-disjoint paths mode in interconnection networks

1994 ◽  
pp. 288-300 ◽  
Author(s):  
Juraj Hromkovič ◽  
Ralf Klasing ◽  
Elena A. Stöhr
Keyword(s):  
Interconnection Networks ◽  
Disjoint Paths ◽  
Vertex Disjoint

scholarly journals Vertex-Disjoint Paths in a 3-Ary n-Cube with Faulty Vertices

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Xiaolei Ma ◽  
Shiying Wang
Keyword(s):  
Interconnection Networks ◽  
Transmission Rate ◽  
Research Topic ◽  
Disjoint Paths ◽  
Important Research ◽  
Important Research Topic ◽  
Vertex Disjoint

The construction of vertex-disjoint paths (disjoint paths) is an important research topic in various kinds of interconnection networks, which can improve the transmission rate and reliability. The k-ary n-cube is a family of popular networks. In this paper, we determine that there are m2≤m≤n disjoint paths in 3-ary n-cube covering Qn3−F from S to T (many-to-many) with F≤2n−2m and from s to T (one-to-many) with F≤2n−m−1 where s is in a fault-free cycle of length three.

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.

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.

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).

The Hyper-Zagreb Index and Some Properties of Graphs

2020 ◽  
pp. 120-134 ◽  
Author(s):  
Rao Li
Keyword(s):  
Topological Index ◽  
Sufficient Conditions ◽  
Disjoint Paths ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Second Zagreb Index ◽  
Vertex Disjoint ◽  
Hamiltonian Properties

Let G = (V(G), E(G)) be a graph. The complement of G is denoted by Gc. The forgotten topological index of G, denoted F(G), is defined as the sum of the cubes of the degrees of all the vertices in G. The second Zagreb index of G, denoted M2(G), is defined as the sum of the products of the degrees of pairs of adjacent vertices in G. A graph Gisk-Hamiltonian if for all X ⊂V(G) with|X| ≤ k, the subgraph induced byV(G) - Xis Hamiltonian. Clearly, G is 0-Hamiltonian if and only if G is Hamiltonian. A graph Gisk-path-coverableifV(G) can be covered bykor fewer vertex-disjoint paths. Using F(Gc) and M2(Gc), Li obtained several sufficient conditions for Hamiltonian and traceable graphs (Rao Li, Topological Indexes and Some Hamiltonian Properties of Graphs). In this chapter, the author presents sufficient conditions based upon F(Gc) and M2(Gc)for k-Hamiltonian, k-edge-Hamiltonian, k-path-coverable, k-connected, and k-edge-connected graphs.

Gossiping in vertex-disjoint paths mode in d-dimensional grids and planar graphs

1993 ◽  
pp. 200-211 ◽  
Author(s):  
Juraj Hromkovič ◽  
Ralf Klasing ◽  
Elena A. Stöhr ◽  
Hubert Wagener
Keyword(s):  
Planar Graphs ◽  
Disjoint Paths ◽  
Vertex Disjoint

Turán numbers of theta graphs

2020 ◽  
Vol 29 (4) ◽  
pp. 495-507 ◽  
Author(s):  
Boris Bukh ◽  
Michael Tait
Keyword(s):  
Disjoint Paths ◽  
Vertex Disjoint

AbstractThe theta graph ${\Theta _{\ell ,t}}$ consists of two vertices joined by t vertex-disjoint paths, each of length $\ell $ . For fixed odd $\ell $ and large t, we show that the largest graph not containing ${\Theta _{\ell ,t}}$ has at most ${c_\ell }{t^{1 - 1/\ell }}{n^{1 + 1/\ell }}$ edges and that this is tight apart from the value of ${c_\ell }$ .

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