# 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 ${H}_{2}$.

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