MATHEMATICA BOHEMICA, Vol. 127, No. 1, pp. 49-55 (2002)

# On signed edge domination numbers of trees

Bohdan Zelinka, Technical University Liberec, Voronezska 13, 460 01 Liberec 1, Czech Republic, e-mail: bohdan.zelinka@vslib.cz

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 $\{-1,1\}$. 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 $\gamma '_s(G)$. If instead of the closed neighbourhood $N_G[e]$ we use the open neighbourhood $N_G(e)=N_G[e]-\{e\}$, we obtain the definition of the signed edge total domination number $\gamma '_{st}(G)$ of $G$. In this paper these concepts are studied for trees.

The number $\gamma '_s(T)$ is determined for $T$ being a star of a path or a caterpillar. Moreover, also $\gamma '_s(C_n)$ for a circuit of length $n$ is determined. For a tree satisfying a certain condition the inequality $\gamma '_s(T) \ge \gamma '(T)$ is stated. An existence theorem for a tree $T$ with a given number of edges and given signed edge domination number is proved. \endgraf At the end similar results are obtained for $\gamma '_{st}(T)$.

Keywords: tree, signed edge domination number, signed edge total domination number

Classification (MSC2000): 05C69, 05C05

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