Optimal upper bound for proper-min-forced vertices

Determine whether the optimal upper bound on the number of proper-min-forced vertices in a closed-twin-free graph of order n is 2n/3-1 or 2n/3.

Background

The paper proves that every connected, nontrivial, closed-twin-free graph of order n has at most 2n/3 proper-min-forced vertices. It also constructs an infinite family, for orders divisible by three, with 2n/3-1 proper-min-forced vertices. Thus, the known construction is within one vertex of the established upper bound, but the authors do not determine whether the bound can be attained or whether it can be improved to 2n/3-1.

The authors note that a more detailed study of the associated colour graphs might establish the stronger bound, although they state that new ideas may be required if 2n/3-1 is indeed optimal.

References

It is tempting to argue that studying the structure of colour graphs more carefully could show the upper bound $2n/3-1$, but we believe that it might not be enough and new ideas are needed if the optimal upper bound is actually $2n/3-1$. For future work, it would be interesting to know whether $2n/3-1$ or $2n/3$ is the optimal bound.

— On the Vertices That Belong to All Minimum Identifying Codes  (2609.09851 - Junnila et al., 9 Sep 2026) in Section 4, Conclusions