Induced Saturation Number

TL;DR

This paper introduces a new concept of induced saturation in graphs, providing foundational results and exact formulas for specific cases like the path on four vertices, and extends the idea to Boolean formulas.

Contribution

It defines the induced saturation number for graphs, establishes basic properties, and derives an exact formula for the case of a path of length four, extending the concept to Boolean formulas.

Findings

Derived the induced saturation number for P4 as eil(n+1)/3eil for n .

Established general properties of induced saturation in graphs.

Extended the notion to Boolean formulas.

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.