# American Institute of Mathematical Sciences

August & September 2019, 12(4&5): 711-721. doi: 10.3934/dcdss.2019045

## Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs

 1 School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China 2 Faculty of Information Studies, Novo Mesto, Slovenia 3 Institut für Informatik, Freie Universität Berlin, Takustraße, D-4195 Berlin, Germany

* Corresponding author: W. Gao

Received  July 2017 Revised  December 2017 Published  November 2018

Fund Project: The first author is supported by NSFC grant (no. 11401519)

The problem of data transmission in communication network can betransformed into the problem of fractional factor existing in graph theory. Inrecent years, the data transmission problem in the specificnetwork conditions has received a great deal of attention, and itraises new demands to the corresponding mathematical model. Underthis background, many advanced results are presented on fractionalcritical deleted graphs and fractional ID deleted graphs. In thispaper, we determine that $G$ is a fractional
 $(g,f,n',m)$
-critical deleted graph if
 $δ(G)≥\frac{b^{2}(i-1)}{a}+n'+2m$
,
 $n>\frac{(a+b)(i(a+b)+2m-2)+bn'}{a}$
, and
 $|N_{G}(x_{1})\cup N_{G}(x_{2})\cup···\cup N_{G}(x_{i})|≥\frac{b(n+n')}{a+b}$
for any independent subset
 $\{x_{1},x_{2},..., x_{i}\}$
of
 $V(G)$
. Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-
 $(g,f,m)$
-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.
Citation: Wei Gao, Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045
##### References:
 [1] E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Appl. Math. Nonl. Sc., 1 (2016), 123-144. [2] J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5. [3] R. Y. Chang, G. Z. Liu and Y. Zhu, Degree conditions of fractional ID-$k$-factor-critical graphs, Bull. Malays. Math. Sci. Soc., 33 (2010), 355-360. [4] W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. [5] W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Degree conditions for fractional $(g,f,n',m)$-critical deleted graphs and fractional ID-$(g,f,m)$-deleted graphs, Bull. Malays. Math. Sci. Soc., 39 (2016), 315-330. doi: 10.1007/s40840-015-0194-1. [6] W. Gao, Y. Guo and K. Y. Wang, Ontology algorithm using singular value decomposition and applied in multidisciplinary, Cluster Comput., 19 (2016), 2201-2210. [7] W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional $(g, f, n)$-critical graphs, J. Korean Math. Soc., 51 (2014), 55-65. doi: 10.4134/JKMS.2014.51.1.055. [8] W. Gao and W. F. Wang, The fifth geometric arithmetic index of bridge graph and carbon nanocones, J. Differ. Equ. Appl., 23 (2017), 100-109. doi: 10.1080/10236198.2016.1197214. [9] W. Gao and W. F. Wang, The eccentric connectivity polynomial of two classes of nanotubes, Chaos Soliton. Fract., 89 (2016), 290-294. doi: 10.1016/j.chaos.2015.11.035. [10] W. Gao, J. L. G. Guirao and H. L. Wu, Two tight independent set conditions for fractional $(g, f, m)$-deleted graphs systems, Qual. Theory Dyn. Syst., 17 (2018), 231-243. doi: 10.1007/s12346-016-0222-z. [11] J. L. G. Guirao and A. C. J. Luo, New trends in nonlinear dynamics and chaoticity, Nonlinear Dynam., 84 (2016), 1-2. doi: 10.1007/s11071-016-2656-x. [12] S. Z. Zhou, Z. R. Sun and Z. R. Xu, A result on $r$-orthogonal factorizations in digraphs, Eur. J. Combin., 65 (2017), 15-23. doi: 10.1016/j.ejc.2017.05.001. [13] S. Z. Zhou, F. Yang and Z. R. Sun, A neighborhood condition for fractional ID-$[a, b]$-factor-critical graphs, Discuss. Mathe. Graph T., 36 (2016), 409-418. doi: 10.7151/dmgt.1864. [14] S. Z. Zhou, L. Xu and Y. Xu, A sufficient condition for the existence of a $k$-factor excluding a given $r$-factor, Appl. Math. Nonl. Sc., 2 (2017), 13-20. [15] S. Z. Zhou, Some Results about Component Factors in Graphs, RAIRO-Oper. Res., 2017. doi: 10.1051/ro/2017045. [16] S. Z. Zhou, Z. R. Sun and H. Liu, A minimum degree condition for fractional ID-[$a,b$]-factor-critical graphs, Bull. Aust. Math. Soc., 86 (2012), 177-183. doi: 10.1017/S0004972711003467.

show all references

##### References:
 [1] E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Appl. Math. Nonl. Sc., 1 (2016), 123-144. [2] J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5. [3] R. Y. Chang, G. Z. Liu and Y. Zhu, Degree conditions of fractional ID-$k$-factor-critical graphs, Bull. Malays. Math. Sci. Soc., 33 (2010), 355-360. [4] W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. [5] W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Degree conditions for fractional $(g,f,n',m)$-critical deleted graphs and fractional ID-$(g,f,m)$-deleted graphs, Bull. Malays. Math. Sci. Soc., 39 (2016), 315-330. doi: 10.1007/s40840-015-0194-1. [6] W. Gao, Y. Guo and K. Y. Wang, Ontology algorithm using singular value decomposition and applied in multidisciplinary, Cluster Comput., 19 (2016), 2201-2210. [7] W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional $(g, f, n)$-critical graphs, J. Korean Math. Soc., 51 (2014), 55-65. doi: 10.4134/JKMS.2014.51.1.055. [8] W. Gao and W. F. Wang, The fifth geometric arithmetic index of bridge graph and carbon nanocones, J. Differ. Equ. Appl., 23 (2017), 100-109. doi: 10.1080/10236198.2016.1197214. [9] W. Gao and W. F. Wang, The eccentric connectivity polynomial of two classes of nanotubes, Chaos Soliton. Fract., 89 (2016), 290-294. doi: 10.1016/j.chaos.2015.11.035. [10] W. Gao, J. L. G. Guirao and H. L. Wu, Two tight independent set conditions for fractional $(g, f, m)$-deleted graphs systems, Qual. Theory Dyn. Syst., 17 (2018), 231-243. doi: 10.1007/s12346-016-0222-z. [11] J. L. G. Guirao and A. C. J. Luo, New trends in nonlinear dynamics and chaoticity, Nonlinear Dynam., 84 (2016), 1-2. doi: 10.1007/s11071-016-2656-x. [12] S. Z. Zhou, Z. R. Sun and Z. R. Xu, A result on $r$-orthogonal factorizations in digraphs, Eur. J. Combin., 65 (2017), 15-23. doi: 10.1016/j.ejc.2017.05.001. [13] S. Z. Zhou, F. Yang and Z. R. Sun, A neighborhood condition for fractional ID-$[a, b]$-factor-critical graphs, Discuss. Mathe. Graph T., 36 (2016), 409-418. doi: 10.7151/dmgt.1864. [14] S. Z. Zhou, L. Xu and Y. Xu, A sufficient condition for the existence of a $k$-factor excluding a given $r$-factor, Appl. Math. Nonl. Sc., 2 (2017), 13-20. [15] S. Z. Zhou, Some Results about Component Factors in Graphs, RAIRO-Oper. Res., 2017. doi: 10.1051/ro/2017045. [16] S. Z. Zhou, Z. R. Sun and H. Liu, A minimum degree condition for fractional ID-[$a,b$]-factor-critical graphs, Bull. Aust. Math. Soc., 86 (2012), 177-183. doi: 10.1017/S0004972711003467.
 [1] Wei Gao, Juan Luis García Guirao, Mahmoud Abdel-Aty, Wenfei Xi. An independent set degree condition for fractional critical deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 877-886. doi: 10.3934/dcdss.2019058 [2] Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715 [3] Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219 [4] Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615 [5] Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008 [6] Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68. [7] Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17 [8] J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413 [9] John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. [10] Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093 [11] Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261 [12] Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036 [13] Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 [14] Jerim Kim, Bara Kim, Hwa-Sung Kim. G/M/1 type structure of a risk model with general claim sizes in a Markovian environment. Journal of Industrial & Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909 [15] Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial & Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609 [16] Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 239-259. doi: 10.3934/mbe.2007.4.239 [17] Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016 [18] Laurent Imbert, Michael J. Jacobson, Jr.. Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$. Advances in Mathematics of Communications, 2013, 7 (4) : 485-502. doi: 10.3934/amc.2013.7.485 [19] Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139 [20] Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001

2017 Impact Factor: 0.561

Article outline