On Collinear Sets in Straight-Line Drawings

Given a planar graph G on n vertices, let fix (G) denote the maximum k such that any straight line drawing of G with possible edge crossings can be made crossing-free by moving at most n − k vertices to new positions. Let v̄(G) denote the maximum possible number of collinear vertices in a crossingfree straight line drawing of G. In view of the relation fix (G) ≤ v̄(G), we are interested in examples of G with v̄(G) = o(n). For each ǫ > 0, we construct an infinite sequence of graphs with v̄(G) = O(nσ+ǫ), where σ < 0.99 is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. Let S be a set of vertices in a crossing-free straight line drawing, all lying on a line l. We call S free if, after any displacement of the vertices in S along l without violating their mutual order, the drawing can be kept crossingfree and straight line by moving the vertices outside S. Let ṽ(G) denote the largest size of a free collinear set maximized over all drawings of G. Noticing relation fix (G) ≥ √ ṽ(G), we get interested in classes of planar graphs with ṽ(G) = Ω(n). We show that outerplanar graphs form one of such classes and prove this way that fix (G) ≥ √ n/2 for every outerplanar G. This slightly improves the lower bound of √ n/3 by Spillner and Wolff and makes the untangling procedure for outerplanar graphs somewhat simpler.