Exact value of the maximal-outerplanar spread for all orders

Determine whether the minimum spread parameter for maximal outerplanar graphs satisfies \(\mathrm{MOP}(n,2)=\left\lceil(4n+10)/9\right\rceil\) for every \(n\geq14\), including residue classes of \(n\) modulo 18 not resolved by the explicit constructions.

Background

The paper proves the lower bound $\mathrm{MOP}(n,2)\geq\lceil(4n+10)/9\rceil$ and establishes equality for orders $n\equiv2\pmod{18}$. For the remaining residue classes, the authors mention computer-verified constructions in supplementary material but do not include those constructions or a proof in the paper.

The authors only prove an upper bound within an additive constant for every order, namely $\mathrm{MOP}(n,2)\leq\lceil(4n+10)/9\rceil+43$ for $n\geq20$. Thus exact equality for all sufficiently large orders, as suggested by their computational evidence, remains unresolved in the paper.