# 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 Faculty of Information Studies, Novo Mesto, Slovenia 2 Institut für Informatik, Freie Universität Berlin, Takustraße, D-4195 Berlin, Germany

Received  July 2017 Revised  December 2017 Published  November 2018

Addendum: Wei Guo was withdraw from the author list for the article
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: Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045
##### References:
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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. [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.
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