On signed distance-$k$-domination in graphs

Xing, Hua Ming, Sun, Liang, and Chen, Xue-Gang. "On signed distance-$k$-domination in graphs." Czechoslovak Mathematical Journal 56.1 (2006): 229-238. <http://eudml.org/doc/31024>.

@article{Xing2006,
abstract = {The signed distance-$k$-domination number of a graph is a certain variant of the signed domination number. If $v$ is a vertex of a graph $G$, the open $k$-neighborhood of $v$, denoted by $N_k(v)$, is the set $N_k(v)=\lbrace u\mid u\ne v$ and $d(u,v)\le k\rbrace $. $N_k[v]=N_k(v)\cup \lbrace v\rbrace $ is the closed $k$-neighborhood of $v$. A function $f\: V\rightarrow \lbrace -1,1\rbrace $ is a signed distance-$k$-dominating function of $G$, if for every vertex $v\in V$, $f(N_k[v])=\sum _\{u\in N_k[v]\}f(u)\ge 1$. The signed distance-$k$-domination number, denoted by $\gamma _\{k,s\}(G)$, is the minimum weight of a signed distance-$k$-dominating function on $G$. The values of $\gamma _\{2,s\}(G)$ are found for graphs with small diameter, paths, circuits. At the end it is proved that $\gamma _\{2,s\}(T)$ is not bounded from below in general for any tree $T$.},
author = {Xing, Hua Ming, Sun, Liang, Chen, Xue-Gang},
journal = {Czechoslovak Mathematical Journal},
keywords = {signed distance-$k$-domination number; signed distance-$k$-dominating function; signed domination number; signed distance--dominating function; signed domination number},
language = {eng},
number = {1},
pages = {229-238},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On signed distance-$k$-domination in graphs},
url = {http://eudml.org/doc/31024},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Xing, Hua Ming
AU - Sun, Liang
AU - Chen, Xue-Gang
TI - On signed distance-$k$-domination in graphs
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 229
EP - 238
AB - The signed distance-$k$-domination number of a graph is a certain variant of the signed domination number. If $v$ is a vertex of a graph $G$, the open $k$-neighborhood of $v$, denoted by $N_k(v)$, is the set $N_k(v)=\lbrace u\mid u\ne v$ and $d(u,v)\le k\rbrace $. $N_k[v]=N_k(v)\cup \lbrace v\rbrace $ is the closed $k$-neighborhood of $v$. A function $f\: V\rightarrow \lbrace -1,1\rbrace $ is a signed distance-$k$-dominating function of $G$, if for every vertex $v\in V$, $f(N_k[v])=\sum _{u\in N_k[v]}f(u)\ge 1$. The signed distance-$k$-domination number, denoted by $\gamma _{k,s}(G)$, is the minimum weight of a signed distance-$k$-dominating function on $G$. The values of $\gamma _{2,s}(G)$ are found for graphs with small diameter, paths, circuits. At the end it is proved that $\gamma _{2,s}(T)$ is not bounded from below in general for any tree $T$.
LA - eng
KW - signed distance-$k$-domination number; signed distance-$k$-dominating function; signed domination number; signed distance--dominating function; signed domination number
UR - http://eudml.org/doc/31024
ER -