Determine the inducibility of the four-vertex path

Determine the inducibility I(P_4) of the four-vertex path P_4 and identify the extremal host graphs attaining it.

Background

The inducibility problem asks for the maximum number, or limiting density, of induced copies of a fixed graph in an arbitrary host graph. The paper states that inducibility is known for all graphs on at most four vertices except the four-vertex path P_4, for which even a plausible extremal construction is unavailable.

References

The inducibility problem is in general wide open, and very difficult. It has been solved for various small graphs, including all graphs on at most four vertices, except, surprisingly, the $4$-vertex path $P_4$; there is not even a sensible conjecture here.

— The semi-inducibility problem  (2501.09842 - Basit et al., 16 Jan 2025) in Section 2.2, “The inducibility problem”