On The Signed Edge Domination Number of Graphs

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Let γ'ₛ(G) be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n (n ≥ 2), γ'ₛ(G)≥ 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 γ'ₛ(G)≤ -m/6|V(G)|. Also for every two natural numbers m and n, we determine γ'ₛ(Kₘ,ₙ), where Kₘ,ₙ is the complete bipartite graph with part sizes m and n.