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

TL;DR

This paper investigates Rall's $1/2$-conjecture in the domination game, proving it for certain graph families and providing computational evidence supporting its validity.

Contribution

It proves the $1/2$-conjecture for hatted cycles and unicyclic graphs, and identifies additional graph families supporting the conjecture.

Findings

Hatted cycles are $1/2$-graphs

Unicyclic graphs fulfill the $1/2$-conjecture

Computer experiments support the conjecture

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 $\left\lceil \frac{n(G)}{2} \right\rceil$. A traceable graph is a $1/2$-graph if $\gamma_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ 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.