Bounds concerning the alliance number

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 $a \leq c$ and $b \leq c$, 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 $a \leq b \leq c$ 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 $\mathop\{\rm order\}(H_1) \leq \mathop\{\rm order\}(H_2)$, 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},
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 $a \leq c$ and $b \leq c$, 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 $a \leq b \leq c$ 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 $\mathop{\rm order}(H_1) \leq \mathop{\rm order}(H_2)$, 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
UR - http://eudml.org/doc/38101
ER -