## On the signed edge domination number of graphs.(English)Zbl 1186.05089

Summary: Let $$\gamma _s^\prime (G)$$ be the signed edge domination number of $$G$$. In 2006, Xu [B. Xu, “Two classes of edge domination in graphs”., Discrete Appl. Math. 154, No.10, 1541–1546 (2006; Zbl 1091.05055)] conjectured that: for any 2-connected graph $$G$$ of order $$n(n\geq 2), \gamma _s^\prime (G)\geq 1$$. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer $$m$$, there exists an $$m$$-connected graph $$G$$ such that $$\gamma _s^\prime (G)\leq -\frac{m}{6}|V(G)|$$. Also for every two natural numbers $$m$$ and $$n$$, we determine $$\gamma_s^\prime (K_{m,n})$$, where $$K_{m,n}$$ is the complete bipartite graph with part sizes $$m$$ and $$n$$.

### MSC:

 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C22 Signed and weighted graphs

### Citations:

Zbl 0979.05081; Zbl 1062.05116; Zbl 1091.05055
Full Text:

### References:

 [1] Bondy, J. A.; Murty, U. S.R., Graph Theory with Applications (1976), North-Holland · Zbl 1134.05001 [2] Xu, B., Two classes of edge domination in graphs, Discrete Appl. Math., 154, 10, 1541-1546 (2006) · Zbl 1091.05055 [3] Xu, B., On edge domination numbers of graphs, Discrete Math., 294, 3, 311-316 (2005) · Zbl 1062.05116 [4] Xu, B., On signed edge domination numbers of graphs, Disc. Math., 239, 179-189 (2001) · Zbl 0979.05081 [5] Zelinka, B., On signed edge domination numbers of trees, Math. Bohem., 127, 1, 49-55 (2002) · Zbl 0995.05112
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