Rooted Uniform Monotone Minimum Spanning Trees

Abstract

A geometric graph G is xy−monotone if each pair of vertices of G is connected by a xy−monotone path. We study the problem of producing the xy−monotone spanning geometric graph of a point set P that (i) has the minimum cost, where the cost of a geometric graph is the sum of the Euclidean lengths of its edges, and (ii) has the least number of edges, in the cases that the Cartesian System xy is specified or freely selected. Building upon previous results, we easily obtain that the two solutions coincide when the Cartesian System is specified and are both equal to the rectangle of influence graph of P . The rectangle of influence graph of P is the geometric graph with vertex set P such that two points p, q ∈ P are adjacent if and only if the rectangle with corners p and q does not include any other point of P . When the Cartesian System can be freely chosen, we note that the two solutions do not necessarily coincide, however we show that they can both be obtained in O(|P | 3 ) time. We also give a simple 2−approximation algorithm for the problem of computing the spanning geometric graph of a k−rooted point set P , in which each root is connected to all the other points (including the other roots) of P by y−monotone paths, that has the minimum cost. *

Exploring Increasing-Chord Paths and Trees

“…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.…”

“…They also proved that the problem remains NP-hard for trees, and gave polynomial-time algorithms in some restricted settings. Recently, Mastakas and Symvonis [8] showed that given a point set S and a point v ∈ S, one can compute a rooted minimum-cost spanning tree in polynomial time, where each point in S \ {v} is connected to v by a path that satisfies some monotonicity property. They also proved that the existence of a monotone rooted spanning tree in a given geometric graph can be decided in polynomial time, and asked whether the decision problem remains NP-hard also for increasing-chord or self-approaching properties.…”