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 H, but changing any white or black edge to gray results in some realization that does have an induced copy of H. We give some general and basic results and then prove that indsat(n,P_4)=⌈ (n+1)/3⌉ for n≥ 4 where P_4 is the path on 4 vertices. We also show how induced saturation in this setting extends to a natural notion of saturation in the context of general Boolean formulas.