Conjectured upper bound for maximal outerplanar graphs

Prove or disprove that every maximal outerplanar graph $G$ of order $n\ge 7$ satisfies $\beta'(G)\le\lceil n/2\rceil$.

Background

The paper proves the general upper bound β′(G)≤⌈2n/3⌉\beta'(G)\le\lceil 2n/3\rceil for maximal outerplanar graphs of order n≥5n\ge 5. It notes that fans have fault-tolerant metric dimension ⌈n/2⌉\lceil n/2\rceil for n≥7n\ge 7 and conjectures that fans attain the maximum fault-tolerant metric dimension among maximal outerplanar graphs. The unresolved problem is therefore to establish or refute the sharper universal upper bound ⌈n/2⌉\lceil n/2\rceil.

References

To conclude the paper, we conjecture the following upper bound based on the known value of the FTMD of fans. A {\em fan} of order $n$, denoted by $F_n$, $n\ge 5$, is a MOP such that exactly one of the vertices is connected to all the other vertices. It is known that $\beta'(F_n)=\left\lceil \frac n2 \right\rceil$, if $n\ge 7$ (see ). We suspect that the maximum value of the FTMD of a MOP is attained by these graphs.

— On the fault-tolerant metric dimension of maximal outerplanar graphs  (2610.13374 - Hernando et al., 9 Oct 2026) in Section Conclusions, fourth Open problem