Upper bounds for inverse domination in graphs

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 $D,D^{\prime }$ with $|D|=\gamma (G)$ and $|D^{\prime }|\le \alpha (G)$. 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 $\gamma (G)\le 5$ or $|V(G)|\le 16$.

9 pages

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• Upper bounds for inverse domination in graphs