Upper bounds for inverse domination in graphs

• Main Authors: Elliot Krop, Jessica McDonald, Gregory Puleo
• Format: Article
• Language: English
• Published: Georgia Southern University 2021-08-01
• Series: Theory and Applications of Graphs
• Subjects: inverse domination; kulli-sigarkanti
• Online Access: https://digitalcommons.georgiasouthern.edu/tag/vol8/iss2/5

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 $|V(G)|\leq 16$.