On the signed edge domination number of graphs

Let γs′(G) be the signed edge domination number of GG. In 2006, Xu conjectured that: for any 2-connected graph GG of order n(n≥2)n(n≥2), γs′(G)≥1. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer mm, there exists an mm-connected graph GG such that γs′(G)≤−m6|V(G)|. Also for every two natural numbers mm and nn, we determine γs′(Km,n), where Km,nKm,n is the complete bipartite graph with part sizes mm and nn.