Induced saturation number

In this paper, we discuss a generalization of the notion of saturation in graphs in order to deal with induced structures. In particular, we define indsat(n,H), which is the fewest number of gray edges in a trigraph so that no realization of that trigraph has an induced copy of HH, but changing any white or black edge to gray results in some realization that does have an induced copy of HH.We give some general and basic results and then prove that indsat(n,P4)=⌈(n+1)/3⌉ for n≥4n≥4 where P4P4 is the path on 44 vertices. We also show how induced saturation in this setting extends to a natural notion of saturation in the context of general Boolean formulas.