A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by γpr(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{pr}(G)$$\end{document}. Let G be a connected {K1,3,K4-e}\documentclass[12pt]{minimal}
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\begin{document}$$\{K_{1,3}, K_{4}-e\}$$\end{document}-free cubic graph of order n. We show that γpr(G)≤10n+627\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{pr}(G)\le \frac{10n+6}{27}$$\end{document} if G is C4\documentclass[12pt]{minimal}
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\begin{document}$$C_{4}$$\end{document}-free and that γpr(G)≤n3+n+69(⌈34(go+1)⌉+1)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}$$\end{document} if G is {C4,C6,C10,…,C2go}\documentclass[12pt]{minimal}
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\begin{document}$$\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}$$\end{document}-free for an odd integer go≥3\documentclass[12pt]{minimal}
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\begin{document}$$g_o\ge 3$$\end{document}; the extremal graphs are characterized; we also show that if G is a 2 -connected, γpr(G)=n3\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{pr}(G) = \frac{n}{3} $$\end{document}. Furthermore, if G is a connected (2k+1)\documentclass[12pt]{minimal}
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\begin{document}$$(2k+1)$$\end{document}-regular {K1,3,K4-e}\documentclass[12pt]{minimal}
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\begin{document}$$\{K_{1,3}, K_4-e\}$$\end{document}-free graph of order n, then γpr(G)≤nk+1\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{pr}(G)\le \frac{n}{k+1} $$\end{document}, with equality if and only if G=L(F)\documentclass[12pt]{minimal}
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\begin{document}$$G=L(F)$$\end{document}, where F≅K1,2k+2\documentclass[12pt]{minimal}
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\begin{document}$$F\cong K_{1, 2k+2}$$\end{document}, or k is even and F≅Kk+1,k+2\documentclass[12pt]{minimal}
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\begin{document}$$F\cong K_{k+1,k+2}$$\end{document}.
