Theory and Applications of Graphs (Aug 2021)

Upper bounds for inverse domination in graphs

  • Elliot Krop,
  • Jessica McDonald,
  • Gregory Puleo

DOI
https://doi.org/10.20429/tag.2021.080205
Journal volume & issue
Vol. 8, no. 2

Abstract

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In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The \emph{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 $|V(G)|\leq 16$.

Keywords