The inclusion of the three-nucleon forces (3NFs) in textit{ab initio} many-body approaches is a formidable task, due to the computational load implied by the treatment of their matrix elements. For this reason, practical applications have mostly been limited to contributions where 3NFs enter as effective two-nucleon interactions. In this contribution, we derive the algebraic diagrammatic construction (ADC) working equations for a specific Feynman diagram of the self-energy that contains a fully irreducible three-nucleon force. This diagram is expected to be the most important among those previously neglected, because it connects dominant excited intermediate state configurations.
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