Untangling planar graphs from a specified vertex position - Hard cases

Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the vertex set of G into the plane. Let fix(G, π) be the maximum integer k such that there exists a crossing-free redrawing π of G which keeps k vertices fixed, i.e., there exist k vertices v1, . . . , vk of G such that π(vi) = π (vi) for i = 1, . . . , k. Given a set of points X , let fix(G) denote the value of fix(G, π) minimized over π locating the vertices of G on X . The absolute minimum of fix(G, π) is denoted by fix(G). For the wheel graph Wn, we prove that fix X(Wn) ≤ (2 + o(1)) √ n for every X . With a somewhat worse constant factor this is as well true for the fan graph Fn. We inspect also other graphs for which it is known that fix (G) = O( √ n). We also show that the minimum value fix(G) of the parameter fix(G) is always attainable by a collinear X .