### Abstract

The connectivity of a network - the minimum number of nodes whose removal will disconnect the network - is directly related to its reliability and fault tolerability, hence an important indicator of the network's robustness. In this paper, we extend the notion of connectivity by introducing two new kinds of connectivity, called structure connectivity and substructure connectivity, respectively. Let H be a certain particular connected subgraph of G. The H-structure connectivity of graph G, denoted κ(G;H), is the cardinality of a minimal set of subgraphs F={H1',H2',. . .,Hm'} in G, such that every Hi'∈F is isomorphic to H, and F's removal will disconnect G. The H-substructure connectivity of graph G, denoted κ^{s}(G;H), is the cardinality of a minimal set of subgraphs F={J_{1}, J_{2}, . . ., J_{m}}, such that every J_{i}∈F is a connected subgraph of H, and F's removal will disconnect G. In this paper, we will establish both κ(Q_{n};H) and κ^{s}(Q_{n};H) for the hypercube Q_{n} and H∈{K_{1}, K_{1,1}, K_{1,2}, K_{1,3}, C_{4}}.

Original language | English |
---|---|

Pages (from-to) | 97-107 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 634 |

DOIs | |

State | Published - 27 Jun 2016 |

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### Keywords

- Hypercube
- Structure connectivity
- Substructure connectivity

### Cite this

*Theoretical Computer Science*,

*634*, 97-107. https://doi.org/10.1016/j.tcs.2016.04.014

}

*Theoretical Computer Science*, vol. 634, pp. 97-107. https://doi.org/10.1016/j.tcs.2016.04.014

**Structure connectivity and substructure connectivity of hypercubes.** / Lin, Cheng Kuan; Zhang, Lili; Fan, Jianxi; Wang, Dajin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Structure connectivity and substructure connectivity of hypercubes

AU - Lin, Cheng Kuan

AU - Zhang, Lili

AU - Fan, Jianxi

AU - Wang, Dajin

PY - 2016/6/27

Y1 - 2016/6/27

N2 - The connectivity of a network - the minimum number of nodes whose removal will disconnect the network - is directly related to its reliability and fault tolerability, hence an important indicator of the network's robustness. In this paper, we extend the notion of connectivity by introducing two new kinds of connectivity, called structure connectivity and substructure connectivity, respectively. Let H be a certain particular connected subgraph of G. The H-structure connectivity of graph G, denoted κ(G;H), is the cardinality of a minimal set of subgraphs F={H1',H2',. . .,Hm'} in G, such that every Hi'∈F is isomorphic to H, and F's removal will disconnect G. The H-substructure connectivity of graph G, denoted κs(G;H), is the cardinality of a minimal set of subgraphs F={J1, J2, . . ., Jm}, such that every Ji∈F is a connected subgraph of H, and F's removal will disconnect G. In this paper, we will establish both κ(Qn;H) and κs(Qn;H) for the hypercube Qn and H∈{K1, K1,1, K1,2, K1,3, C4}.

AB - The connectivity of a network - the minimum number of nodes whose removal will disconnect the network - is directly related to its reliability and fault tolerability, hence an important indicator of the network's robustness. In this paper, we extend the notion of connectivity by introducing two new kinds of connectivity, called structure connectivity and substructure connectivity, respectively. Let H be a certain particular connected subgraph of G. The H-structure connectivity of graph G, denoted κ(G;H), is the cardinality of a minimal set of subgraphs F={H1',H2',. . .,Hm'} in G, such that every Hi'∈F is isomorphic to H, and F's removal will disconnect G. The H-substructure connectivity of graph G, denoted κs(G;H), is the cardinality of a minimal set of subgraphs F={J1, J2, . . ., Jm}, such that every Ji∈F is a connected subgraph of H, and F's removal will disconnect G. In this paper, we will establish both κ(Qn;H) and κs(Qn;H) for the hypercube Qn and H∈{K1, K1,1, K1,2, K1,3, C4}.

KW - Hypercube

KW - Structure connectivity

KW - Substructure connectivity

UR - http://www.scopus.com/inward/record.url?scp=84966714059&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2016.04.014

DO - 10.1016/j.tcs.2016.04.014

M3 - Article

AN - SCOPUS:84966714059

VL - 634

SP - 97

EP - 107

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -