The vertex bound conjecture for minimal forbidden induced subgraphs of subgraph complementation number

Let $c_2(G)$ denote the minimum cardinality of a subgraph complementation system for a graph $G$, and fix a natural number $k$. A minimal forbidden induced subgraph for the property $c_2(G)\leq k$ is a graph $F$ such that $c_2(F)>k$ but every proper induced subgraph $F-v$ satisfies $c_2(F-v)\leq k$. Vertex bound conjecture. Every minimal forbidden induced subgraph for the property $c_2(G)\leq k$ has at most $2k+2$ vertices.

The preceding theorem establishes that the class of graphs with $c_2(G)\leq k$ is characterized by finitely many forbidden induced subgraphs, but does not give an explicit bound on their orders. This conjecture proposes a linear bound based on results concerning linear forests.