  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.
