Convex Drawings of Graphs in Two and Three Dimensions (Preliminary Version)

We provideO(n)-timealgorithmsfor constructingthe following types of drawings of n-vertex 3-connectedplanar graphs: 2D convex grid drawings with (3n) (3n=2) area under the edge L1-resolution rule; 2D strictly convex grid drawings with O(n 3) O(n 3) area under the edge resolution rule; 2D strictly convex drawings with O(1) O(n) area under the vertex-resolution rule, and with vertex coordinates represented by O(n logn)-bit rational numbers; 3D convex drawings with O(1) O(1) O(n) volume under the vertex-resolution rule, and with vertex coordinates represented by O(n logn)-bit rational numbers. We also show the following lower bounds: For in nitely many n-vertex graphs G, if G has a straightline 2D convex drawing in a w h grid satisfying the edge L1-resolution rule then w; h 5n=6 + (1) and w + h 8n=3+ (1). For in nitely many bounded-degree triconnected planar graphs G with n vertices, any 3D convex drawing of G must have volume 2 (n) under the angular resolution rule.

• Type: Conference
• Year: 1996
• Where: COMPGEOM
• Authors: Marek Chrobak, Michael T. Goodrich, Roberto Tamassia
• Keywords: COMPGEOM 1996, Convex Drawing, Convex Grid Drawings, Discrete Geometry, Edge L1-resolution Rule