Improvement of the outerplanar scramble-number bound

Determine whether the bound $sn(G)\leq 3\sqrt{2n}+8$ for outerplanar graphs can be improved, or construct a family of outerplanar graphs whose scramble number attains the order of the bound in Theorem 3.1.

Background

The main theorem establishes the explicit upper bound $sn(G)\leq 3\sqrt{2n}+8$ for every outerplanar graph on n vertices. The leading constant arises from balancing the size of a potential hitting set against the sizes of cuts used to separate components. Because previously studied planar families have scramble number roughly proportional to √n with a smaller apparent constant, the paper leaves open whether its outerplanar bound is quantitatively improvable or is asymptotically sharp for some family.