Structural characterization of the n−6 discrepancy bound

Characterize the structural graph properties that force the minimal zero forcing set discrepancy md(G) to be at most n−6 for graphs of order n≥17.

Background

The main theorem establishes the numerical upper bound md(G)≤n−6 for graphs of order at least 17, but the proof proceeds through case analyses based primarily on the zero forcing number and forcing-chain structure. The authors leave unresolved a more intrinsic structural explanation for why larger discrepancies cannot occur.

References

What structural characteristics limit $md(G)$ to $n-6$ for graphs of order $n\geq17$?

— Maximizing the discrepancy between zero forcing parameters relative to graph order  (2609.35522 - McKay et al., 28 Sep 2026) in Section 6, Conclusion and Open Problems

However, we do suspect that the same result could be proven for $n\geq7$. Thus, we provide a conjecture but leave the question open.

\begin{conjecture} If $G$ is a graph of order $n\geq 7$, then $md(G) \leq n-6$. \end{conjecture}

— Maximizing the discrepancy between zero forcing parameters relative to graph order  (2609.35522 - McKay et al., 28 Sep 2026) in Section 6, Conclusion and Open Problems

Do these bounds on $md(G)$ and $pd(G)$ extend to skew forcing? To positive semidefinite forcing?

— Maximizing the discrepancy between zero forcing parameters relative to graph order  (2609.35522 - McKay et al., 28 Sep 2026) in Section 6, Conclusion and Open Problems