Classification of cubic graphs violating the domination comparison

Determine whether every connected cubic graph G satisfying gamma_{1,2}(G) > c_ind(G) can be obtained from a trilobite or from a graph H(k) by a bounded local modification, or whether the number of sporadic exceptions grows without bound.

Background

The paper disproves the conjecture that a connected cubic graph G has gamma_{1,2}(G) > c_ind(G) if and only if G is a trilobite. It identifies additional counterexamples at orders 18, 20, and 22 and constructs an infinite family H(k) of non-trilobite counterexamples.

The authors observe that the class of graphs violating gamma_{1,2}(G) <= c_ind(G) appears richer than the trilobites and grows with n. The unresolved question asks whether all such graphs admit a bounded-local-modification description based on trilobites and H(k), or whether sporadic exceptions proliferate without bound.

References

Is every connected cubic graph $G$ with $\gamma_{1,2}(G)>c_{\mathrm{ind}(G)$ obtained from a trilobite or from a member of $H(k)$ by a bounded local modification, or does the number of ``sporadic'' exceptions (such as the remaining examples at $n=18,20,22$) grow without bound?

— 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)