## On signed edge domination numbers of trees.(English)Zbl 0995.05112

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 $$\{-1,1\}$$. If $$\sum_{x\in N[e]} f(x)\geq 1$$ for each $$e\in E(G)$$, then $$f$$ is called a signed edge dominating functions on $$G$$. The minimum of the values $$\sum _{x\in E(G)} f(x)$$, taken over all signed edge dominating functions $$f$$ 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. Also $$\gamma_s'(C_n)$$ for a circuit of length $$n$$ is determined. For a tree satisfying a certain condition the inequality $$\gamma_s'(T) \geq \gamma'(T)$$ is stated. An existence theorem for a tree $$T$$ with a given number of edges and given signed edge domination number is proved. At the end similar results are obtained for $$\gamma_{st}'(T)$$.

### MSC:

 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C05 Trees
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