scholarly journals Vertex-disjoint path covers in graphs

2017 ◽  
Author(s):  
Renzo Gómez
Keyword(s):  
Disjoint Path ◽  
Vertex Disjoint

Seja G um grafo conexo e P um conjunto de caminhos disjuntos nos vértices em G. Dizemos que P é uma cobertura por caminhos se cada vértice de G pertence a algum caminho em P . No problema da cobertura mínima por caminhos, o objetivo é encontrar uma cobertura com o menor número de caminhos. Nesse problema, que é sabido ser NP-difícil, o conjunto P pode conter caminhos triviais. Estudamos uma variante desse problema onde o objetivo é encontrar uma cobertura sem caminhos triviais. Usando a decomposição de Edmonds-Gallai, mostramos que o problema de decidir se um grafo tem tal cobertura pode ser reduzido a um problema de emparelhamento em um grafo bipartido. Além disso, mostramos resultados de inaproximabilidade para ambos os problemas de cobertura: com e sem caminhos triviais.

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 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].

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 On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

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