Untangling planar graphs from a specified vertex position—Hard cases

AbstractGiven 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 fixX(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 fixX(Wn)≤(2+o(1))n for every X. With a somewhat worse constant factor this is also 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 fixX(G) is always attainable by a collinear X