The g-Good-Neighbor Conditional Diagnosability of Locally Twisted Cubes

Wei, Yulong; Xu, Min

doi:10.1007/s40305-017-0166-2

Cited by 17 publications

(4 citation statements)

References 32 publications

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“…A connected graph G is said to be g-good-neighbor connected if G has a g-good-neighbor cut. For more research on g-good-neighbor connectivity, we refer to [16,18,22,24,28,29,33,32,36,38,39].…”

Section: The G-good-neighbor Connectivitymentioning

confidence: 99%

On the g-good-neighbor connectivity of graphs

Wang

Mao

Hsieh

et al. 2020

Theoretical Computer Science

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Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network G. In 1996, Fàbrega and Fiol proposed the g-good-neighbor connectivity of G. In this paper, we show that 1 ≤ κ g (G) ≤ n − 2g − 2 for 0 ≤ g ≤ ∆(G), n−3 2 , and graphs with κ g (G) = 1, 2 and trees with κ g (T n ) = n − t for 4 ≤ t ≤ n+2 2 are characterized, respectively. In the end, we get the three extremal results for the g-good-neighbor connectivity. The g-extra (edge-)connectivityFàbrega and Fiol [8,9] introduced the following measures. A subset of vertices S is said to be a cutset if G − S is not connected. A cutset S is called an R g -cutset, where g is a non-negative integer, if every component of G − S has at least g + 1 vertices. If G has at least one R g -cutset, the g-extra connectivity of G, denoted by κ g (G), is then defined as the minimum cardinality over all R g -cutsets of G. A connected graph G is said to be g-extra connected if G has a g-extra cut.Given a graph and a non-negative integer g, the g-extra edge-connectivity, written as λ g (G), is the minimal cardinality of a set of edges in G, if it exists, whose deletion disconnects G and leaves each remaining component with more than g vertices.Note that κ 0 (G) = κ(G) and λ 0 (G) = λ(G) for any graph if is not a complete graph. Therefore, the g-extra connectivity (resp. g-extra edge connectivity) can be regarded as a generalization of the classical connectivity (resp. classical edge connectivity) that provides measures that are more accurate for reliability and fault tolerance for large-scale parallel processing systems. Regarding the computational complexity of the problem, based on thorough research, no polynomial-time algorithm has been presented to compute κ g for a general graph; nor has there been any tight upper bound for κ g [6]. However, λ 1 can be computed by solving numerous network flow problems [7]. The g-good-neighbor connectivityLet G = (V, E) be connected. A fault set F ⊆ V is called a g-good-neighbor faulty set if |N (v)∩(V −F )| ≥ g for every vertex v in V − F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G − F is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the ggood-neighbor connectivity of G, denoted by κ g (G). A connected graph G is said to be g-good-neighbor

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Section: The G-good-neighbor Connectivitymentioning

confidence: 99%

On the g-good-neighbor connectivity of graphs

Wang

Mao

Hsieh

et al. 2020

Theoretical Computer Science

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“…Furthermore, the fault-tolerance or reliability of a network are important. A network is usually represented by a graph where vertices represent processors and edges respresent communication links between [3]. The fault-tolerance or reliability of a network are often measured by the connectivity of a correasponding graph.…”

Section: Introductionmentioning

confidence: 99%

The g-extra connectivity of the Mycielskian

Li¹,

Zhang²,

Ye³

2020

Preprint

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The g-extra connectivity is an important parameter to measure the ability of tolerance and reliability of interconnection networks. Given a connected graph G = (V, E) and a non-disconnected and every component of G − S has at least g + 1 vertices. The cardinality of the minimum g-extra cut is defined as the g-extra connectivity of G, denoted by κ g (G). In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph µ(G), which is called the Mycielskian of G. This paper investigates the relationship of the g-extra connectivity of the Mycielskian µ(G) and the graph G, moreover, show that κ 2g+1 (µ(G)) = 2κ g (G) + 1 for g ≥ 1 and κ g (G) ≤ min{g + 1, ⌊ n 2 ⌋}.

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“…The MM * model which is the development of the MM model [9], proposed by Sengupta and Dahbura [12], assumes that each processor has to test two processors if the processor is adjacent to the latter two processors. Some references related to fault diagnosis studies under the PMC model or MM * model can be seen in [2,5,6,[13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%