Inducibility of the four-vertex path

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

Background

The inducibility problem asks for the maximum number of induced copies of a fixed graph in an n-vertex host graph and its limiting density. The paper notes that inducibility is known for graphs on at most four vertices except for 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”