# 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. Google Scholar [2] J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5. Google Scholar [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. Google Scholar [4] W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] S. Z. Zhou, Some Results about Component Factors in Graphs, RAIRO-Oper. Res., 2017. doi: 10.1051/ro/2017045. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [2] J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5. Google Scholar [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. Google Scholar [4] W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] S. Z. Zhou, Some Results about Component Factors in Graphs, RAIRO-Oper. Res., 2017. doi: 10.1051/ro/2017045. Google Scholar [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. Google Scholar
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