Let G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} be a simple graph. A function f:V⟶{-1,1}\documentclass[12pt]{minimal}
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\begin{document}$$f:V\longrightarrow \{-1,1\}$$\end{document} is a signed dominating function if for every vertex v∈V\documentclass[12pt]{minimal}
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\begin{document}$$v \in V$$\end{document}, the closed neighborhood of v contains more vertices with function value 1 than with -1\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document}. The weight of a function f is ω(f)=∑v∈Vf(v)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (f)= \sum _{v \in V} f(v)$$\end{document}. The signed domination number of G, γs(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _s(G)$$\end{document}, is the minimum weight of a signed dominating function on G. A signed dominating function of weight γs(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _s(G)$$\end{document} is called γs(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _s(G)$$\end{document}-function. A γs(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _s(G)$$\end{document}-function can also be represented by a set of ordered pairs Sf={(v,f(v)):v∈V}\documentclass[12pt]{minimal}
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\begin{document}$$S_f=\{(v, f(v)) :v\in V\}$$\end{document}. A subset T of Sf\documentclass[12pt]{minimal}
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\begin{document}$$S_f$$\end{document} is called a forcing subset of Sf\documentclass[12pt]{minimal}
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\begin{document}$$S_f$$\end{document} if Sf\documentclass[12pt]{minimal}
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\begin{document}$$S_f$$\end{document} is the unique extension of T to a γs(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _s(G)$$\end{document}-function. The forcing signed domination number of Sf\documentclass[12pt]{minimal}
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\begin{document}$$S_f$$\end{document}f(Sf,γs)=min{|T|:T\documentclass[12pt]{minimal}
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\begin{document}$$f(S_f, \gamma _s) = {\rm min} \{|T|: T$$\end{document} is a forcing subset of Sf}\documentclass[12pt]{minimal}
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\begin{document}$$S_f\}$$\end{document} and the forcing signed domination number of G, f(G,γs)\documentclass[12pt]{minimal}
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\begin{document}$$f(G,\gamma _s)$$\end{document}, is defined by f(G,γs)=min{f(Sf,γs):Sf\documentclass[12pt]{minimal}
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\begin{document}$$f(G,\gamma _s) = {\rm min} \{f(S_f,\gamma _s): S_f$$\end{document} is a γs(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _s(G)$$\end{document}-function}\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}. In this paper, we deal with the signed domination number of several classes of graph. Also the forcing signed domination number of some graphs are determined.
