Two-thirds upper bound for (1,2)-domination in triangle-free cubic graphs

Determine whether every triangle-free cubic graph G satisfies \(\gamma_{1,2}(G)\le \frac23|V(G)|\).

Background

The $(1,2)$-domination number is the minimum size of a dominating set in which every selected vertex has at least two neighbors within the selected set. The paper discusses a proposed universal upper bound for triangle-free cubic graphs.

The authors show that the Goldberg family, which is triangle-free for $k\ge4$, satisfies the proposed bound strictly using the exact formula established in the paper. They do not settle the question for all triangle-free cubic graphs.