Let G=(V

E) be a graph

a function f:V→{－1

1} is said to be a cycle signed domination function (CSDF) of G if  ∑v

∈V(C)

f(v)≥1 holds for any induced cycle C of G

where the cycle signed domination number of G  is defined as γ

sc

(G)=min{∑

v∈V f(v)

f  is a CSDF of G}.Some lower bounds of the cycle signed domination number of a graph are obtained

and all graphs G with δ≥2 and γ

sc

(G)=4－|V(G)| are characterized.