On Rall's $1/2$-conjecture on the domination game

Abstract

The $1/2$-conjecture on the domination game asserts that if $G$ is a traceable graph, then the game domination number ${\gamma }_g(G)$ of $G$ is at most $\lceil \frac{n(G)}{2}\rceil$. A traceable graph is a $1/2$-graph if ${\gamma }_g(G)=\lceil \frac{n(G)}{2}\rceil$ holds. It is proved that the so-called hatted cycles are $1/2$-graphs and that unicyclic graphs fulfill the $1/2$-conjecture. Several additional families of graphs that support the conjecture are determined and computer experiments related to the conjecture described.