On the g-good-neighbor connectivity of graphs

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, Fabrega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq \kappa^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{\Delta(G),\left\lfloor \frac{n-3}{2}\right\rfloor\right\}$, and graphs with $\kappa^g(G)=1,2$ and trees with $\kappa^g(T_n)=n-t$ for $4\leq t\leq \frac{n+2}{2}$ are characterized, respectively. In the end, we get the three extremal results for the $g$-good-neighbor connectivity.

[1]  Miguel Angel Fiol,et al.  Extraconnectivity of graphs with large girth , 1994, Discret. Math..

[2]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  Sun-Yuan Hsieh,et al.  Conditional Diagnosability of $(n,k)$ -Star Graphs Under the PMC Model , 2018, IEEE Transactions on Dependable and Secure Computing.

[5]  Eddie Cheng,et al.  A general approach to deriving the g-good-neighbor conditional diagnosability of interconnection networks , 2019, Theor. Comput. Sci..

[6]  Junming Xu.  Topological Structure and Analysis of Interconnection Networks , 2002, Network Theory and Applications.

[7]  Abdol-Hossein Esfahanian,et al.  Generalized Measures of Fault Tolerance with Application to N-Cube Networks , 1989, IEEE Trans. Computers.

[8]  Miguel Angel Fiol,et al.  On the extraconnectivity of graphs , 1996, Discret. Math..

[9]  Min Xu,et al.  The 1, 2-good-neighbor conditional diagnosabilities of regular graphs , 2018, Appl. Math. Comput..

[10]  Shahram Latifi,et al.  Conditional Connectivity Measures for Large Multiprocessor Systems , 1994, IEEE Trans. Computers.

[11]  Bernd Krauskopf,et al.  On the 1 , 2009 .

[12]  Jia Guo,et al.  Conditional diagnosability of the SPn graphs under the comparison diagnosis model , 2018, Appl. Math. Comput..

[13]  Li Xu,et al.  The Extra Connectivity and Conditional Diagnosability of Alternating Group Networks , 2015, IEEE Transactions on Parallel and Distributed Systems.

[14]  Mei Lu,et al.  The g-good-neighbor conditional diagnosability of star graphs under the PMC and MM* model , 2017, Theor. Comput. Sci..

[15]  Jianxi Fan,et al.  Diagnosability of the exchanged crossed cube , 2018, Int. J. Comput. Math. Comput. Syst. Theory.

[16]  Shuming Zhou,et al.  Conditional diagnosability and strong diagnosability of shuffle-cubes under the comparison model , 2015, Int. J. Comput. Math..

[17]  Miroslaw Malek,et al.  A comparison connection assignment for diagnosis of multiprocessor systems , 1980, ISCA '80.

[18]  Shuming Zhou,et al.  The g-good-neighbor diagnosability of (n, k)-star graphs , 2017, Theor. Comput. Sci..

[19]  Min Xu,et al.  On g-good-neighbor conditional diagnosability of (n, k)-star networks , 2017, Theor. Comput. Sci..

[20]  Shuming Zhou,et al.  Fault tolerance and diagnosability of burnt pancake networks under the comparison model , 2015, Theor. Comput. Sci..

[21]  S. Louis Hakimi,et al.  On Computing a Conditional Edge-Connectivity of a Graph , 1988, Inf. Process. Lett..

[22]  Eddie Cheng,et al.  Structural Properties and Fault Resiliency of Interconnection Networks , 2019, From Parallel to Emergent Computing.

[23]  Di Wu,et al.  Some extremal results on the colorful monochromatic vertex-connectivity of a graph , 2015, J. Comb. Optim..

[24]  Zhenhua Wang,et al.  The 2-good-neighbor connectivity and 2-good-neighbor diagnosability of bubble-sort star graph networks , 2017, Discret. Appl. Math..

[25]  Jun-Ming Xu,et al.  Fault-tolerance of (n, k)-star networks , 2014, Appl. Math. Comput..

[26]  Cheng-Kuan Lin,et al.  The g-good-neighbor conditional diagnosability of hypercube under PMC model , 2012, Appl. Math. Comput..

[27]  Shiying Wang,et al.  The g-good-neighbor diagnosability of locally twisted cubes , 2017, Theor. Comput. Sci..

[28]  Min Xu,et al.  The g-Good-Neighbor Conditional Diagnosability of Locally Twisted Cubes , 2017, ArXiv.

[29]  Xiao Qin,et al.  The ${\schmi g}$-Good-Neighbor Conditional Diagnosability of ${\schmi k}$-Ary ${\schmi n}$-Cubes under the PMC Modeland MM* Model , 2015, IEEE Transactions on Parallel and Distributed Systems.

[30]  Li Xu,et al.  The g-Good-Neighbor Conditional Diagnosability of Arrangement Graphs , 2016, IEEE Trans. Dependable Secur. Comput..

[31]  Aixia Liu,et al.  g-Good-neighbor conditional diagnosability measures for 3-ary n-cube networks , 2016, Theor. Comput. Sci..

[32]  Sun-Yuan Hsieh,et al.  Conditional Diagnosability of k-Ary n-Cubes under the PMC Model , 2012, TODE.

[33]  Jimmy J. M. Tan,et al.  Conditional diagnosability measures for large multiprocessor systems , 2005, IEEE Transactions on Computers.

[34]  Jia Guo,et al.  Conditional diagnosability of the round matching composition networks , 2017, Theor. Comput. Sci..

[35]  Jie Wu,et al.  Fault Tolerance Measures for m-Ary n-Dimensional Hypercubes Based on Forbidden Faulty Sets , 1998, IEEE Trans. Computers.

[36]  Gerald M. Masson,et al.  An 0(n2.5) Fault Identification Algorithm for Diagnosable Systems , 1984, IEEE Transactions on Computers.

[37]  Ingo Schiermeyer.  On minimally rainbow k-connected graphs , 2013, Discret. Appl. Math..

[38]  Mujiangshan Wang,et al.  The g-good-neighbor and g-extra diagnosability of networks , 2019, Theor. Comput. Sci..

[39]  Jun-Ming Xu,et al.  On fault tolerance of (n, k)-star networks , 2017, Theor. Comput. Sci..

[40]  Jun-Ming Xu,et al.  On Super and Restricted Connectivity of Some Interconnection Networks , 2010, Ars Comb..

[41]  Hui Jiang,et al.  Erdős-Gallai-Type Results for Total Monochromatic Connection of Graphs , 2019, Discuss. Math. Graph Theory.

[42]  GERNOT METZE,et al.  On the Connection Assignment Problem of Diagnosable Systems , 1967, IEEE Trans. Electron. Comput..

[43]  Abhijit Sengupta,et al.  On self-diagnosable multiprocessor systems: diagnosis by the comparison approach , 1989, [1989] The Nineteenth International Symposium on Fault-Tolerant Computing. Digest of Papers.