Counterexamples separating strict and non-strict multi-k-11-representation

Construct counterexamples, for k equal to 0 or 1, showing that the strict and non-strict multi-k-11-representation numbers are not equivalent.

Background

For k at least 2, both strict and non-strict multi-k-11-representation numbers equal 1 because every graph is k-11-representable. The unresolved issue concerns k=0 and k=1, where the two notions may differ; the authors specifically ask whether counterexamples exist.

References

The notions of strict and non-strict multi-k-11-representation numbers are equivalent for k ≥ 2. What can be said about k ∈ 0, 1? Is it possible to construct any counterexamples in this case?

— On 1-11-representability and multi-1-11-representability of graphs  (2501.13871 - Alshammari et al., 23 Jan 2025) in Section 5, Concluding remarks, page 9