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August & September
2019, 12(4&5): 1179-1185.
doi: 10.3934/dcdss.2019081
Vulnerability of super connected split graphs and bisplit graphs
| 1. | School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China |
| 2. | School of Science, Jimei University, Xiamen, Fujian 361021, China |
A graph $G = (C, I, E)$ is called a split graph if its vertex set $V$ can be partitioned into a clique $C$ and an independent set $I$. A graph $G = (Y \cup Z, I, E)$ is called a bisplit graph if its vertex set $V$ can be partitioned into three stable sets $I, Y,Z$ such that $Y \cup Z$ induces a complete bipartite graph and an independent set $I$. A connected graph $G$ is called supper-$κ$ (resp. super-$λ$) if every minimum vertex cut (edge cut) of $G$ is the set of neighbors of some vertex (the edges of incident to some vertex) in $G$. In this note, we show that: split graphs and bisplit graphs are super-$κ$ and super-$λ$.
Mathematics Subject Classification:
Primary: 05C40; Secondary: 05C90.
Citation:
Litao Guo, Bernard L. S. Lin. Vulnerability of super connected split graphs and bisplit graphs. Discrete & Continuous Dynamical Systems - S,
2019, 12
(4&5)
: 1179-1185.
doi: 10.3934/dcdss.2019081
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References:
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Graph Theory and Its Application, Academic Press, New York, 1976. |
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L. T. Guo, W. Yang and X., F. Guo,
Super-connectivity of Kronecker products of split graphs, powers of cycles, powers of paths and complete graphs, Applied Mathematics Letters, 26 (2013), 120-123.
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M. Metsidik and E. Vumar,
Edge vulnerability parameters of bisplit graphs, Computers and Mathematics with Applications, 56 (2008), 1741-1747.
doi: 10.1016/j.camwa.2008.04.015. |
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G. J. Woeginger,
The toughness of split graphs, Discrete Math., 190 (1998), 295-297.
doi: 10.1016/S0012-365X(98)00156-3. |
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S. Zhang, Q. Zhang and H. Yang,
Vulnerability parameters of split graphs, Int. J. Comput. Math., 85 (2008), 19-23.
doi: 10.1080/00207160701365721. |
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Q. Zhang and S. Zhang,
Edge vulnerability parameters of split graphs, Applied Mathematics Letters, 19 (2006), 916-920.
doi: 10.1016/j.aml.2005.09.011. |
show all references
References:
| [1] |
J. A. Bondy and U. S. R. Murty,
Graph Theory and Its Application, Academic Press, New York, 1976. |
| [2] |
A. Brandst |
| [3] |
S. F |
| [4] |
L. T. Guo, C. Qin and X. F. Guo,
Super connectivity of Kronecker products of graphs, Information Processing Letters, 110 (2010), 659-661.
doi: 10.1016/j.ipl.2010.05.013. |
| [5] |
L. T. Guo, W. Yang and X., F. Guo,
Super-connectivity of Kronecker products of split graphs, powers of cycles, powers of paths and complete graphs, Applied Mathematics Letters, 26 (2013), 120-123.
doi: 10.1016/j.aml.2012.04.006. |
| [6] |
M. Metsidik and E. Vumar,
Edge vulnerability parameters of bisplit graphs, Computers and Mathematics with Applications, 56 (2008), 1741-1747.
doi: 10.1016/j.camwa.2008.04.015. |
| [7] |
G. J. Woeginger,
The toughness of split graphs, Discrete Math., 190 (1998), 295-297.
doi: 10.1016/S0012-365X(98)00156-3. |
| [8] |
S. Zhang, Q. Zhang and H. Yang,
Vulnerability parameters of split graphs, Int. J. Comput. Math., 85 (2008), 19-23.
doi: 10.1080/00207160701365721. |
| [9] |
Q. Zhang and S. Zhang,
Edge vulnerability parameters of split graphs, Applied Mathematics Letters, 19 (2006), 916-920.
doi: 10.1016/j.aml.2005.09.011. |
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