This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required

A proximity drawing of a graph is one in which pairs of adjacent vertices are drawn relatively close together according to some proximity measure while pairs of non-adjacent vertices are drawn relatively far apart. The fundamental question concerning proximity drawability is: Given a graph G and a definition of proximity, is it possible to construct a proximity drawing of G? We consider this question for outerplanar graphs with respect to an infinite family of proximity drawings called fi-drawings. These drawings include as special cases the well-known Gabriel drawings (when fi = 1), and relative neighborhood drawings (when fi = 2). We first show that all biconnected outerplanar graphs are fi-drawable for all values of fi such that 1 fi 2. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer [20, 22], that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanar graphs that do not admit any...

We provide O(n)-time algorithms for constructing the following types of drawings of n-vertex 3-connected planar graphs: ffl 2D convex grid drawings with (3n) \Theta (3n=2) area under the edge L1 -resolution rule; ffl 2D strictly convex grid drawings with O(n 3 ) \Theta O(n 3 ) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) \Theta O(n) area under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers; ffl 3D convex drawings with O(1) \Theta O(1) \Theta O(n) volume under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers. We also show the following lower bounds: ffl For infinitely many n-vertex graphs G, if G has a straightline 2D convex drawing in a w \Theta h grid satisfying the edge L1 -resolution rule then w;h 5n=6 +\Omega\Gamma20 and w + h 8n=3 + \Omega\Gamma838 ffl For infinitely many bounded-degree triconnected planar graphs G with n ver...

This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required.