2013/06/23 by Soroush Alamdari, Timothy M. Chan, Alamdari, Soroush +7 · 2 citations
Computer Science · #Advanced Graph Theory Research #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.1306.5460
openalex publication_date 2013/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner. We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3) constructing a self-approaching Steiner network connecting a given set of points. We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in R2 and R3, but it is NP-hard to test if a given graph drawing in R3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.