Three-Dimensional Grid Drawings with Sub-quadratic Volume
Dujmovic, Vida and Wood, David R. (2004) Three-Dimensional Grid Drawings with Sub-quadratic Volume. In: Graph Drawing 11th International Symposium, GD 2003, September 21-24, 2003, Perugia, Italy , pp. 190-201 (Official URL: http://dx.doi.org/10.1007/978-3-540-24595-7_18).
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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(m^2). These results (partially) solve open problems due to Pach, Thiele, and Tóth [Graph Drawing 1997] and Felsner, Liotta, and Wismath [Graph Drawing 2001].
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