Tight untangling bounds for cubic triconnected planar graphs

Determine tight asymptotic bounds on fix_{𝒞₃}(n) for the class of n-vertex cubic triconnected planar graphs, where fix(G) is the maximum number of vertices that can be kept fixed while untangling a straight-line drawing of G.

Background

Linear free sets are known for cubic triconnected planar graphs, which imply Ω(n^{1/2}) untangling bounds via free-set arguments. However, a matching upper bound is not known.

Establishing tight bounds would complement known results for trees and outerplanar graphs, where untangling bounds are tight, and clarify the effect of triconnectivity and regularity on untangling complexity.