Many-to-many disjoint paths in faulty hypercubes

Chen, Xie-Bin

doi:10.1016/j.ins.2009.05.006

Cited by 58 publications

(31 citation statements)

References 23 publications

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“…The disjoint path cover problem has been studied for several classes of graphs: hypercubes [6,9,14,16], recursive circulants [18,19,25,26], and hypercube-like graphs [25,26]. The structure of the cubes of connected graphs was investigated with respect to single-source 3-disjoint path covers [24].…”

Section: Disjoint Path Coversmentioning

confidence: 99%

Disjoint path covers in cubes of connected graphs

Park

Ihm

2014

Discrete Mathematics

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Given a graph G, and two vertex sets S and T of size k each, a many-tomany k-disjoint path cover of G joining S and T is a collection of k disjoint paths between S and T that cover every vertex of G. It is classified as paired if each vertex of S must be joined to a designated vertex of T , or unpaired if there is no such constraint. In this article, we first present a necessary and sufficient condition for the cube of a connected graph to have a paired 2-disjoint path cover. Then, a corresponding condition for the unpaired type of 2-disjoint path cover problem is immediately derived. It is also shown that these results can easily be extended to determine if the cube of a connected graph has a hamiltonian path from a given vertex to another vertex that passes through a prescribed edge.

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Section: Disjoint Path Coversmentioning

confidence: 99%

Disjoint path covers in cubes of connected graphs

Park

Ihm

2014

Discrete Mathematics

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“…It is also deeply related to the full utilization of nodes in interconnection networks [22]. Several special classes of graphs have been studied in terms of the existence of desired disjoint path covers: hypercubes [6,8,13], recursive circulants [16,22,23], and hypercube-like graphs [22,23]. The many-tomany 2-disjoint path cover problem with respect to the cubes of connected graphs was also studied recently [21].…”

Section: Introductionmentioning

confidence: 99%

Single-source three-disjoint path covers in cubes of connected graphs

Park

Ihm

2013

Information Processing Letters

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A k-disjoint path cover (k-DPC for short) of a graph is a set of k internally vertex-disjoint paths from given sources to sinks that collectively cover every vertex in the graph. In this paper, we establish a necessary and sufficient condition for the cube of a connected graph to have a 3-DPC joining a single source to three sinks. We also show that the cube of a connected graph always has a 3-DPC joining arbitrary two vertices.

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“…When a graph contains faulty elements, whether vertices or edges, its k-disjoint path cover naturally means a k-disjoint path cover of the graph with the faulty elements deleted. Some works on the construction of k-DPC's in hypercubes [3,4,7,11] and hypercube-like graphs [15,29,30] can be found. The embedding of a linear array or a ring into an interconnection network can be modeled as finding a long path or cycle, possibly a Hamiltonian path or cycle.…”

Section: Introductionmentioning

confidence: 99%

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

Jo¹,

Park²,

Chwa³

2013

Information Sciences

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A paired many-to-many k-disjoint path cover (paired k-DPC for short) of a graph is a set of k vertex-disjoint paths joining k distinct source-sink pairs that altogether cover every vertex of the graph. We consider the problem of constructing paired 2-DPC's in an m-dimensional bipartite HL-graph, X m , and its application in finding the longest possible paths. It is proved that every X m , m ≥ 4, has a fault-free paired 2-DPC if there are at most m − 3 faulty edges and the set of sources and sinks is balanced in the sense that it contains the same number of vertices from each part of the bipartition. Furthermore, every X m , m ≥ 4, has a paired 2-DPC in which the two paths have the same length if each source-sink pair is balanced. Using 2-DPC properties, we show that every X m , m ≥ 3, with either at most m − 2 faulty edges or one faulty vertex and at most m − 3 faulty edges is strongly Hamiltonian-laceable.

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Many-to-many disjoint paths in faulty hypercubes

Cited by 58 publications

References 23 publications

Disjoint path covers in cubes of connected graphs

Disjoint path covers in cubes of connected graphs

Single-source three-disjoint path covers in cubes of connected graphs

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

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