# Upper bounds for inverse domination in graphs

## TL;DR

This paper investigates bounds related to the inverse domination conjecture in graphs, proving it under certain relaxed conditions and for specific graph classes, advancing understanding of dominating set structures.

## Contribution

The paper proves the inverse domination conjecture with a relaxed upper bound and confirms its validity for graphs with small domination number or limited size.

## Key findings

- The conjecture holds if the upper bound is frac{3}{2}alpha(G) - 1 for non-clique graphs.
- The conjecture is true for graphs with domination number leq 5.
- It also holds for graphs with up to 16 vertices.

## Abstract

In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The Inverse Domination Conjecture says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with $|D|=\gamma(G)$ and $|D'| \leq \alpha(G)$. Here we prove that this statement is true if the upper bound $\alpha(G)$ is replaced by $\frac{3}{2}\alpha(G) - 1$ (and $G$ is not a clique). We also prove that the conjecture holds whenever $\gamma(G)\leq 5$ or $|V(G)|\leq 16$.