Three-Dimensional Grid Drawings with Sub-quadratic Volume

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A O(n3/2) volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n2). These results (partially) solve open problems due to Pach, Thiele, and T´oth (1997) and Felsner, Liotta, and Wismath (2001).

• Type: Conference
• Year: 2003
• Where: GD
• Authors: Vida Dujmovic, David R. Wood
• Keywords: Bounded Complete, GD 2003, Integer Coordinates, Three-dimensional Grid Drawings