On signed edge domination numbers of trees

Zelinka, Bohdan. "On signed edge domination numbers of trees." Mathematica Bohemica 127.1 (2002): 49-55. <http://eudml.org/doc/249027>.

@article{Zelinka2002,
abstract = {The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having a common end vertex with $e$. Let $f$ be a mapping of the edge set $E(G)$ of $G$ into the set $\lbrace -1,1\rbrace $. If $\sum _\{x\in N[e]\} f(x)\ge 1$ for each $e\in E(G)$, then $f$ is called a signed edge dominating function on $G$. The minimum of the values $\sum _\{x\in E(G)\} f(x)$, taken over all signed edge dominating function $f$ on $G$, is called the signed edge domination number of $G$ and is denoted by $\gamma ^\{\prime \}_s(G)$. If instead of the closed neighbourhood $N_G[e]$ we use the open neighbourhood $N_G(e)=N_G[e]-\lbrace e\rbrace $, we obtain the definition of the signed edge total domination number $\gamma ^\{\prime \}_\{st\}(G)$ of $G$. In this paper these concepts are studied for trees. The number $\gamma ^\{\prime \}_s(T)$ is determined for $T$ being a star of a path or a caterpillar. Moreover, also $\gamma ^\{\prime \}_s(C_n)$ for a circuit of length $n$ is determined. For a tree satisfying a certain condition the inequality $\gamma ^\{\prime \}_s(T) \ge \gamma ^\{\prime \}(T)$ is stated. An existence theorem for a tree $T$ with a given number of edges and given signed edge domination number is proved. At the end similar results are obtained for $\gamma ^\{\prime \}_\{st\}(T)$.},
author = {Zelinka, Bohdan},
journal = {Mathematica Bohemica},
keywords = {tree; signed edge domination number; signed edge total domination number; tree; signed edge domination number; signed edge total domination number},
language = {eng},
number = {1},
pages = {49-55},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On signed edge domination numbers of trees},
url = {http://eudml.org/doc/249027},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Zelinka, Bohdan
TI - On signed edge domination numbers of trees
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 1
SP - 49
EP - 55
AB - The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having a common end vertex with $e$. Let $f$ be a mapping of the edge set $E(G)$ of $G$ into the set $\lbrace -1,1\rbrace $. If $\sum _{x\in N[e]} f(x)\ge 1$ for each $e\in E(G)$, then $f$ is called a signed edge dominating function on $G$. The minimum of the values $\sum _{x\in E(G)} f(x)$, taken over all signed edge dominating function $f$ on $G$, is called the signed edge domination number of $G$ and is denoted by $\gamma ^{\prime }_s(G)$. If instead of the closed neighbourhood $N_G[e]$ we use the open neighbourhood $N_G(e)=N_G[e]-\lbrace e\rbrace $, we obtain the definition of the signed edge total domination number $\gamma ^{\prime }_{st}(G)$ of $G$. In this paper these concepts are studied for trees. The number $\gamma ^{\prime }_s(T)$ is determined for $T$ being a star of a path or a caterpillar. Moreover, also $\gamma ^{\prime }_s(C_n)$ for a circuit of length $n$ is determined. For a tree satisfying a certain condition the inequality $\gamma ^{\prime }_s(T) \ge \gamma ^{\prime }(T)$ is stated. An existence theorem for a tree $T$ with a given number of edges and given signed edge domination number is proved. At the end similar results are obtained for $\gamma ^{\prime }_{st}(T)$.
LA - eng
KW - tree; signed edge domination number; signed edge total domination number; tree; signed edge domination number; signed edge total domination number
UR - http://eudml.org/doc/249027
ER -