Finite-threshold strengthening of the failed domination conjecture

Determine whether there exists a finite threshold t such that every cubic graph G with c_ind(G) >= n/2 + t satisfies gamma_{1,2}(G) <= c_ind(G).

Background

The paper disproves the proposed threshold n/2+2 in the conjecture of Erveš and Tepeh: its family H(k) satisfies c_ind(H(k)) = n/2+2 and gamma_{1,2}(H(k)) = n/2+3. Computational examples of order 22 even attain c_ind(G) = n/2+3, so any repaired threshold would have to satisfy t >= 4.

The authors leave unresolved whether increasing the threshold by any finite amount can make the implication true. This is a concrete question about whether a universal finite threshold separates the induced-cycle parameter from the (1,2)-domination number.

References

Is there a finite threshold $t$ for which the statement of Conjecture~\ref{Con_ET2} becomes true after replacing $n/2+2$ by $n/2+t$? Our examples at $n=22$ already reach $c_{\mathrm{ind}(G)=n/2+3,$ so $t\geq4$ would be required. It is not clear that any finite $t$ suffices.

— Counterexamples to two conjectures on (1, 2)-domination in cubic graphs  (2608.17851 - Knor et al., 18 Aug 2026) in Problem environment, Section 4 (Concluding remarks)