Multiscale planar graph generation

Multiscale Planar Graph Generation

The study of network representations of physical, biological, and social phenomena can help us better understand the structural and functional dynamics of their networks and formulate predictive models of these phenomena. However, due to the scarcity of real-world network data owing to factors such as cost and effort required in collection of network data and the sensitivity of this data toward...

Planar Graph Growth Constants

Steven Finch August 25, 2004 A graph of order  consists of a set of  vertices (points) together with a set of edges (unordered pairs of distinct points). Note that loops and multiple parallel edges are automatically disallowed. Two vertices joined by an edge are called adjacent. Two graphs  and  are isomorphic if there is a one-to-one map from the vertices of  to the vertices of  that pre...

Morphing Planar Graph Drawings

The study of planar graphs dates back to Euler and the earliest days of graph theory. Centuries later came the proofs by Wagner, Fáry and Stein that every planar graph can be drawn with straight line segments for the edges, and the algorithm by Tutte for constructing such straight-line drawings given in his 1963 paper, “How to Draw a Graph”. With more recent attention to complexity issues, this...

The Random Planar Graph

Untangling a Planar Graph

A straight-line drawing δ of a planar graph G need not be plane, but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G, δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G, δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-...