# Bounds concerning the alliance number

Mathematica Bohemica (2009)

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

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## Abstract

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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 $a\left(G\right)$, strong defensive alliance number $\stackrel{^}{a}\left(G\right)$, and global defensive alliance number ${\gamma }_{a}\left(G\right)$. In this paper, we consider relationships between these parameters and the domination number $\gamma \left(G\right)$. For any positive integers $a,b,$ and $c$ satisfying $a\le c$ and $b\le c$, there is a graph $G$ with $a=a\left(G\right)$, $b=\gamma \left(G\right)$, and $c={\gamma }_{a}\left(G\right)$. 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 $\gamma \left(G\right)=a$, ${\gamma }_{a}\left(G\right)=b$, and ${\gamma }_{\stackrel{^}{a}}\left(G\right)=c$. Given two connected graphs ${H}_{1}$ and ${H}_{2}$, where $\mathrm{order}\left({H}_{1}\right)\le \mathrm{order}\left({H}_{2}\right)$, there exists a graph $G$ with a unique minimum defensive alliance isomorphic to ${H}_{1}$ and a unique minimum strong defensive alliance isomorphic to ${H}_{2}$.

## How to cite

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Bullington, 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 - 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 -

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