On the minus domination number of graphs

Liu, Hailong, and Sun, Liang. "On the minus domination number of graphs." Czechoslovak Mathematical Journal 54.4 (2004): 883-887. <http://eudml.org/doc/30907>.

@article{Liu2004,
abstract = {Let $G = (V,E)$ be a simple graph. A $3$-valued function $f\:V(G)\rightarrow \lbrace -1,0,1\rbrace $ is said to be a minus dominating function if for every vertex $v\in V$, $f(N[v]) = \sum _\{u\in N[v]\}f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$. The weight of a minus dominating function $f$ on $G$ is $f(V) = \sum _\{v\in V\}f(v)$. The minus domination number of a graph $G$, denoted by $\gamma ^-(G)$, equals the minimum weight of a minus dominating function on $G$. In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$, then \[ \gamma ^-(G)\ge 4\bigl (\sqrt\{n + 1\}-1\bigr )-n. \] (2) For any negative integer $k$ and any positive integer $m\ge 3$, there exists a graph $G$ with girth $m$ such that $\gamma ^-(G)\le k$. Therefore, two open problems about minus domination number are solved.},
author = {Liu, Hailong, Sun, Liang},
journal = {Czechoslovak Mathematical Journal},
keywords = {minus dominating function; minus domination number; minus dominating function; minus domination number},
language = {eng},
number = {4},
pages = {883-887},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the minus domination number of graphs},
url = {http://eudml.org/doc/30907},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Liu, Hailong
AU - Sun, Liang
TI - On the minus domination number of graphs
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 4
SP - 883
EP - 887
AB - Let $G = (V,E)$ be a simple graph. A $3$-valued function $f\:V(G)\rightarrow \lbrace -1,0,1\rbrace $ is said to be a minus dominating function if for every vertex $v\in V$, $f(N[v]) = \sum _{u\in N[v]}f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$. The weight of a minus dominating function $f$ on $G$ is $f(V) = \sum _{v\in V}f(v)$. The minus domination number of a graph $G$, denoted by $\gamma ^-(G)$, equals the minimum weight of a minus dominating function on $G$. In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$, then \[ \gamma ^-(G)\ge 4\bigl (\sqrt{n + 1}-1\bigr )-n. \] (2) For any negative integer $k$ and any positive integer $m\ge 3$, there exists a graph $G$ with girth $m$ such that $\gamma ^-(G)\le k$. Therefore, two open problems about minus domination number are solved.
LA - eng
KW - minus dominating function; minus domination number; minus dominating function; minus domination number
UR - http://eudml.org/doc/30907
ER -