# Article

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Keywords:
tree; signed edge domination number; signed edge total domination number
Summary:
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 $\gamma ^{\prime }_{st}(T)$.
References:
[1] E. Xu: On signed domination numbers of graphs. Discr. Math. (submitted).
[2] T. W. Haynes, S. T. Hedetniemi, P. J. Slater: Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998. MR 1605684

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