We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a represen...

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...

We construct an in nite planar graph that contains every planar graph as a minor.

Approximation algorithms for a class of planar graph problems, including planar independent set, planar vertex cover and planar dominating set, were intensively studied. The current upper bound on the running time of the polynomial time approximation schemes (PTAS) for these planar graph problems is of 2 n. Here we study the lower bound on the running time of the PTAS for these planar graph pro...

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 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...

In JM94] we used a branch and cut algorithm in order to determine a maximum weight planar subgraph of a given graph. One of the motivations was to produce a nice drawing of a given graph by drawing the found maximum planar subgraph, and then augmenting this drawing by the removed edges. Our experiments indicate that drawing algorithms for planar graphs which require 2-or 3-connectivity, resp. d...

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-...