A Toughness Condition for Fractional (k, m)-deleted Graphs Revisited

In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional (k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional (k, m)-deleted graphs improving the existing one. Finally, we state an open problem.

[1]  Frank Harary,et al.  Graph Theory , 2016 .

[2]  Jiahua Jin,et al.  Multiple solutions of the Kirchhoff-type problem in RN , 2016 .

[3]  Jeffrey G. Andrews,et al.  Guest editorial: geometry and random graphs for the analysis and design of wireless networks , 2009, IEEE Journal on Selected Areas in Communications.

[4]  Weifan Wang,et al.  New isolated toughness condition for fractional $(g,f,n)$-critical graphs , 2017 .

[5]  Jeffrey G. Andrews,et al.  Stochastic geometry and random graphs for the analysis and design of wireless networks , 2009, IEEE Journal on Selected Areas in Communications.

[6]  Mir Saman Pishvaee,et al.  A graph theoretic-based heuristic algorithm for responsive supply chain network design with direct and indirect shipment , 2011, Adv. Eng. Softw..

[7]  Ali Haghighi,et al.  A Graph Portioning Approach for Hydraulic Analysis-Design of Looped Pipe Networks , 2015, Water Resources Management.

[8]  Veena R. Desai,et al.  (β ,α)−Connectivity Index of Graphs , 2017 .

[9]  Wei Gao,et al.  The eccentric connectivity polynomial of two classes of nanotubes , 2016 .

[10]  M. Lorini,et al.  Applying Graph Theory to Design Networks of Protected Areas: Using Inter-Patch Distance for Regional Conservation Planning , 2011 .

[11]  Joaquim F. Martins-Filho,et al.  New Graph Model to Design Optical Networks , 2015, IEEE Communications Letters.

[12]  Leomar S. da Rosa,et al.  Graph-Based Transistor Network Generation Method for Supergate Design , 2016, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[13]  EXPONENTIAL LOWER BOUNDS FOR THE FBI TRANSFORM ON C k 0 ( R n ) , 2007 .

[14]  Fu Lin,et al.  Design of Optimal Sparse Interconnection Graphs for Synchronization of Oscillator Networks , 2013, IEEE Transactions on Automatic Control.

[15]  Guizhen Liu,et al.  Toughness and the existence of fractional k-factors of graphs , 2008, Discret. Math..

[16]  Peter Ashwin,et al.  On designing heteroclinic networks from graphs , 2013, 1302.0984.

[17]  Francesco Archetti,et al.  Graph models and mathematical programming in biochemical network analysis and metabolic engineering design , 2008, Comput. Math. Appl..

[18]  Wei Gao,et al.  Ontology algorithm using singular value decomposition and applied in multidisciplinary , 2016, Cluster Computing.

[19]  Lan Xu,et al.  A sufficient condition for the existence of a k-factor excluding a given r-factor , 2017 .

[20]  Wei Gao,et al.  The fifth geometric-arithmetic index of bridge graph and carbon nanocones , 2017 .

[21]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[22]  Vasek Chvátal,et al.  Tough graphs and hamiltonian circuits , 1973, Discret. Math..

[23]  Juan Luis García Guirao,et al.  New trends in nonlinear dynamics and chaoticity , 2016 .

[24]  Hui Ye,et al.  A toughness condition for fractional (k, m)-deleted graphs , 2013, Inf. Process. Lett..

[25]  G. Rizzelli,et al.  Impairment-aware design of translucent DWDM networks based on the k-path connectivity graph , 2012, IEEE/OSA Journal of Optical Communications and Networking.

[26]  Sizhong Zhou.  A NEIGHBOURHOOD CONDITION FOR GRAPHS TO BE FRACTIONAL (k, m)-DELETED GRAPHS* , 2009, Glasgow Mathematical Journal.