In this paper, we investigate dynamic modal logics with global and local dynamic operations, which update the accessibility relation of a graph model. We introduce the hybrid logic \(\textsf{GLV}\) of global link variations, which involves dynamic operators of global link cutting, adding and rotating simultaneously. We provide the Hilbert-style calculus \(\mathsf {C_{GLV}}\) and prove that it is sound and strongly complete with respect to \(\textsf{GLV}\) by constructing a family of canonical models inductively. We study the hybrid extensions of the logic \(\textsf{LLD}\) of definable link deletion introduced by Li (2020). We provide a sound and complete tableau calculus \(\mathcal {T}(\textsf{LLD}(@))\) for the logic \(\textsf{LLD}(@)\) . Then we extend \(\textsf{LLD}(@)\) to the logics \(\textsf{LLV}(@,X)\) of local link variations and provide them with sound and complete tableau calculi. Furthermore, we extend the logic \(\textsf{LLV}\) to \(\textsf{LLV}(\downarrow \hspace{-.3em}\, ,\textsf{E})\) by adding the hybrid operator \(\downarrow \hspace{-.3em}a.\) and existential modality \(\textsf{E}\) . By defining local named dynamic operators and providing recursion axioms for them, we obtain a sound and strongly complete calculus \(\mathsf {C_{LLV}}\) for \(\textsf{LLV}(\downarrow \hspace{-.3em}\, ,\textsf{E})\) . Finally, we show that for any set X of global or local operators, the calculus \(\mathsf {C_{GLV}}(X)\) and \(\mathsf {C_{LLV}}(X)\) are still sound and strongly complete w.r.t the logic \(\textsf{GLV}(X)\) and \(\textsf{LLV}(\downarrow \hspace{-.3em}\, ,\textsf{E},X)\) , respectively.
