A subset S of vertex set V(G) of a graph G is a zero forcing set of G if iteratively adding vertices to S from V(G)\S\documentclass[12pt]{minimal}
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\begin{document}$$V(G){\setminus }S$$\end{document} that are the unique neighbor in V(G)\S\documentclass[12pt]{minimal}
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\begin{document}$$V(G){\setminus } S$$\end{document} of some vertex in S, results in the entire V(G) of G. Additionally, if the subgraph induced by S is connected, then S is a connected forcing set of G. The zero (resp., connected) forcing number, denoted by F(G) (resp., Fc(G)\documentclass[12pt]{minimal}
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\begin{document}$$F_c(G)$$\end{document}), of G is the minimum cardinality of a zero (resp., connected) forcing set of G. Davila and Kenter [Theory Appl. Graphs, 2(2) (2015) Article 1] proved that F(G)≤n-g+2\documentclass[12pt]{minimal}
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\begin{document}$$F(G)\le n-g+2$$\end{document} for graphs G of finite girth g and order n(≥g)\documentclass[12pt]{minimal}
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\begin{document}$$n (\ge g)$$\end{document}. In this paper, first, we restrict G to be connected and improve the upper bound according to the value of g: F(G)≤n-g+2\documentclass[12pt]{minimal}
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\begin{document}$$F(G)\le n-g+2$$\end{document} when g=3,4\documentclass[12pt]{minimal}
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\begin{document}$$g=3, 4$$\end{document} or n; F(G)≤n-g+1\documentclass[12pt]{minimal}
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\begin{document}$$F(G)\le n-g+1$$\end{document} when g=5,6\documentclass[12pt]{minimal}
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\begin{document}$$g=5, 6$$\end{document} or n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document}(n≥6)\documentclass[12pt]{minimal}
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\begin{document}$$(n \ge 6)$$\end{document} and F(G)≤n-g\documentclass[12pt]{minimal}
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\begin{document}$$F(G)\le n-g$$\end{document} when 7≤g≤n-2\documentclass[12pt]{minimal}
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\begin{document}$$7\le g\le n-2$$\end{document}. Further, the extremal graphs are characterized, respectively. Davila et al. [Graphs Combin. 34 (2018) 1159-1174] proved a similar upper bound on Fc(G)\documentclass[12pt]{minimal}
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\begin{document}$$F_c(G)$$\end{document} for 2-connected graphs G with n vertices and finite girth g: Fc(G)≤n-g+2\documentclass[12pt]{minimal}
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\begin{document}$$F_c(G)\le n-g+2$$\end{document}. Secondly, we prove that Fc(G)=n-g+2\documentclass[12pt]{minimal}
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\begin{document}$$F_c(G)=n-g+2$$\end{document} if and only if G is Cn\documentclass[12pt]{minimal}
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\begin{document}$$C_{n}$$\end{document}, or Kn\documentclass[12pt]{minimal}
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\begin{document}$$K_{n}$$\end{document}, or Ka,b\documentclass[12pt]{minimal}
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\begin{document}$$K_{a,b}$$\end{document} for integers a,b≥2\documentclass[12pt]{minimal}
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\begin{document}$$a, b \ge 2$$\end{document}, a+b=n\documentclass[12pt]{minimal}
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\begin{document}$$a+b=n$$\end{document}. Finally, we also characterize all connected triangle-free or subcubic graphs G of order n with F(G)=n-3\documentclass[12pt]{minimal}
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\begin{document}$$F(G)=n-3$$\end{document}.
