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Motivated by Fan's result, we say that an induced subgraph $H$ of $G$ is $f_1$-heavy if for every pair of vertices $u,v\\in V(H)$, $d_{H}(u,v)=2$ implies $\\max\\{d(u),d(v)\\}\\geq (n+1)/2$. For a given graph $R$, $G$ is called $R$-$f_1$-heavy if every induced subgraph of $G$ isomorphic to $R$ is $f_1$-heavy. 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In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a 2-connected graph to be Hamiltonian. Motivated by Fan's result, we say that an induced subgraph $H$ of $G$ is $f_1$-heavy if for every pair of vertices $u,v\\in V(H)$, $d_{H}(u,v)=2$ implies $\\max\\{d(u),d(v)\\}\\geq (n+1)/2$. For a given graph $R$, $G$ is called $R$-$f_1$-heavy if every induced subgraph of $G$ isomorphic to $R$ is $f_1$-heavy. 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