# Bounds concerning the alliance number

Grady Bullington; Linda Eroh; Steven J. Winters

Mathematica Bohemica (2009)

- Volume: 134, Issue: 4, page 387-398
- ISSN: 0862-7959

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topBullington, Grady, Eroh, Linda, and Winters, Steven J.. "Bounds concerning the alliance number." Mathematica Bohemica 134.4 (2009): 387-398. <http://eudml.org/doc/38101>.

@article{Bullington2009,

abstract = {P. Kristiansen, S. M. Hedetniemi, and S. T. Hedetniemi, in Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177, and T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, in Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), introduced the defensive alliance number $a(G)$, strong defensive alliance number $\hat\{a\}(G)$, and global defensive alliance number $\gamma _a(G)$. In this paper, we consider relationships between these parameters and the domination number $\gamma (G)$. For any positive integers $a,b,$ and $c$ satisfying $a \le c$ and $b \le c$, there is a graph $G$ with $a=a(G)$, $b=\gamma (G)$, and $c=\gamma _a(G)$. For any positive integers $a,b,$ and $c$, provided $a \le b \le c$ and $c$ is not too much larger than $a$ and $b$, there is a graph $G$ with $\gamma (G)=a$, $\gamma _a(G)=b$, and $\gamma _\{\hat\{a\}\}(G)=c$. Given two connected graphs $H_1$ and $H_2$, where $\mathop \{\rm order\}(H_1) \le \mathop \{\rm order\}(H_2)$, there exists a graph $G$ with a unique minimum defensive alliance isomorphic to $H_1$ and a unique minimum strong defensive alliance isomorphic to $H_2$.},

author = {Bullington, Grady, Eroh, Linda, Winters, Steven J.},

journal = {Mathematica Bohemica},

keywords = {defensive alliance; global defensive alliance; domination number; defensive alliance; global defensive alliance; domination number},

language = {eng},

number = {4},

pages = {387-398},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Bounds concerning the alliance number},

url = {http://eudml.org/doc/38101},

volume = {134},

year = {2009},

}

TY - JOUR

AU - Bullington, Grady

AU - Eroh, Linda

AU - Winters, Steven J.

TI - Bounds concerning the alliance number

JO - Mathematica Bohemica

PY - 2009

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 134

IS - 4

SP - 387

EP - 398

AB - P. Kristiansen, S. M. Hedetniemi, and S. T. Hedetniemi, in Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177, and T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, in Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), introduced the defensive alliance number $a(G)$, strong defensive alliance number $\hat{a}(G)$, and global defensive alliance number $\gamma _a(G)$. In this paper, we consider relationships between these parameters and the domination number $\gamma (G)$. For any positive integers $a,b,$ and $c$ satisfying $a \le c$ and $b \le c$, there is a graph $G$ with $a=a(G)$, $b=\gamma (G)$, and $c=\gamma _a(G)$. For any positive integers $a,b,$ and $c$, provided $a \le b \le c$ and $c$ is not too much larger than $a$ and $b$, there is a graph $G$ with $\gamma (G)=a$, $\gamma _a(G)=b$, and $\gamma _{\hat{a}}(G)=c$. Given two connected graphs $H_1$ and $H_2$, where $\mathop {\rm order}(H_1) \le \mathop {\rm order}(H_2)$, there exists a graph $G$ with a unique minimum defensive alliance isomorphic to $H_1$ and a unique minimum strong defensive alliance isomorphic to $H_2$.

LA - eng

KW - defensive alliance; global defensive alliance; domination number; defensive alliance; global defensive alliance; domination number

UR - http://eudml.org/doc/38101

ER -

## References

top- Kristiansen, P., Hedetniemi, S. M., Hedetniemi, S. T., Alliances in graphs, J. Comb. Math. Comb. Comput. 48 (2004), 157-177. (2004) Zbl1064.05112MR2036749
- Haynes, T. W., Hedetniemi, S. T., Henning, M. A., 10.37236/1740, Electron. J. Comb. 10 (2003), # R47. (2003) Zbl1031.05096MR2026883DOI10.37236/1740

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