Induced Saturation Number

Abstract: 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 ${\rm indsat}(n,H)$, 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 ${\rm indsat}(n,P_4)=\lceil (n+1)/3\rceil$ for $n\geq 4$ 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.