Exploring Increasing-Chord Paths and Trees

A straight-line drawing Γ of a graph G = (V,E) is a drawing of G in the Euclidean plane, where every vertex in G is mapped to a distinct point, and every edge in G is mapped to a straight line segment between their endpoints. A path P in Γ is called increasing-chord if for every four points (not necessarily vertices) a, b, c, d on P in this order, the Euclidean distance between b, c is at most the Euclidean distance between a, d. A spanning tree T rooted at some vertex r in Γ is called increasing-chord if T contains an increasing-chord path from r to every vertex in T . We prove that given a vertex r in a straight-line drawing Γ, it is NP-complete to decide whether Γ contains an increasing-chord spanning tree rooted at r, which answers a question posed by Mastakas and Symvonis [8]. We also shed light on the problem of finding an increasing-chord path between a pair of vertices in Γ, but the computational complexity question remains open.