Textbooks often defines a one-particle irreducible diagram (1PI diagram) as a connected diagram which does not fall into two pieces if you cut one internal line. Is this internal line the full (connected) propagator or the free propagator?
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TL;DR: It's the bare/free propagator.
If there are no tadpoles, then cutting full (connected) propagators vs. bare/free propagators formally amounts to the same notion of one-particle irreducible$^\dagger$ (1PI) since we can rewrite a full (connected) propagator $$G_c~=~G_0\sum_{n=0}^{\infty}(\Sigma G_0)^n$$ as a geometric series of bare/free propagators $G_0$, and vice-versa $$G_0~=~G\sum_{n=0}^{\infty}(-\Sigma G_c)^n.$$ Here $$\Sigma~=~G_0^{-1}-G_c^{-1} $$ is the self-energy, which is 1PI if there are no tadpoles, cf. my Phys.SE answer here.
$^\dagger$ 1PI is called 2-edge-connected by mathematicians. [Let us mention for completeness that arXiv:2202.12296 defines a one-vertex-irreducible (1VI) graph as a connected graph that cannot be disconnected by removal of a single vertex.]
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