For two graphs A and B, a graph G is called {A,B}\documentclass[12pt]{minimal}
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\begin{document}$$\{A,B\}$$\end{document}-free if G contains neither A nor B as an induced subgraph. Let Pn\documentclass[12pt]{minimal}
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\begin{document}$$P_{n}$$\end{document} denote the path of order n. For nonnegative integers k, ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} and m, let Nk,ℓ,m\documentclass[12pt]{minimal}
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\begin{document}$$N_{k,\ell ,m}$$\end{document} be the graph obtained from K3\documentclass[12pt]{minimal}
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\begin{document}$$K_{3}$$\end{document} and three vertex-disjoint paths Pk+1\documentclass[12pt]{minimal}
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\begin{document}$$P_{k+1}$$\end{document}, Pℓ+1\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ell +1}$$\end{document}, Pm+1\documentclass[12pt]{minimal}
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\begin{document}$$P_{m+1}$$\end{document} by identifying each of the vertices of K3\documentclass[12pt]{minimal}
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\begin{document}$$K_{3}$$\end{document} with one endvertex of one of the paths. Let Zk=Nk,0,0\documentclass[12pt]{minimal}
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\begin{document}$$Z_{k}=N_{k,0,0}$$\end{document} and Bk,ℓ=Nk,ℓ,0\documentclass[12pt]{minimal}
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\begin{document}$$B_{k,\ell }=N_{k,\ell ,0}$$\end{document}. Bedrossian characterized all pairs {A,B}\documentclass[12pt]{minimal}
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\begin{document}$$\{A,B\}$$\end{document} of connected graphs such that every 2-connected {A,B}\documentclass[12pt]{minimal}
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\begin{document}$$\{A,B\}$$\end{document}-free graph is Hamiltonian. All pairs appearing in the characterization involve the claw (K1,3\documentclass[12pt]{minimal}
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\begin{document}$$K_{1,3}$$\end{document}) and one of N1,1,1\documentclass[12pt]{minimal}
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\begin{document}$$N_{1,1,1}$$\end{document}, P6\documentclass[12pt]{minimal}
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\begin{document}$$P_{6}$$\end{document} and B1,2\documentclass[12pt]{minimal}
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\begin{document}$$B_{1,2}$$\end{document}. In this paper, we characterize connected graphs that are (i) {K1,3,Z2}\documentclass[12pt]{minimal}
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\begin{document}$$\{K_{1,3},Z_{2}\}$$\end{document}-free but not B1,1\documentclass[12pt]{minimal}
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\begin{document}$$B_{1,1}$$\end{document}-free, (ii) {K1,3,B1,1}\documentclass[12pt]{minimal}
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\begin{document}$$\{K_{1,3},B_{1,1}\}$$\end{document}-free but not P5\documentclass[12pt]{minimal}
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\begin{document}$$P_{5}$$\end{document}-free, or (iii) {K1,3,B1,2}\documentclass[12pt]{minimal}
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\begin{document}$$\{K_{1,3},B_{1,2}\}$$\end{document}-free but not P6\documentclass[12pt]{minimal}
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\begin{document}$$P_{6}$$\end{document}-free. The third result is closely related to Bedrossian’s characterization. Furthermore, we apply our characterizations to some forbidden pair problems.
