Induced saturation number

In this paper, we discuss the induced saturation number. It is a nice generalization of the saturation number that will allow us to consider induced subgraphs. We define the induced saturation number, indsat(n,H), to be the fewest number of gray edges in a trigraph T such that H does not appear in any realization of T , but if a black or white edge of T is flipped to gray then there exists a realization of T with H as an induced subgraph. We will provide some general results and prove that for a path on four vertices, indsat(n,P4) = ⌈n+1 3 ⌉ for n ≥ 4. We will also discuss the injective coloring number and a generalization of that.