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(n 3/2 ) 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(n 2 ). These results (partially) solve open problems due to Pach, Thiele, and Toth (Graph Drawing 1997) and Fel- sner, Liotta, and Wismath (Graph Drawing 2001).
['Vida Dujmović', 'David R. Wood']
Three-Dimensional Grid Drawings with Sub-quadratic Volume