It is shown that for every finite set of disjoint convex polygonal obstacles in the plane, with a total of n vertices, the free space around the obstacles can be partitioned into open convex cells whose dual graph (defined below) is 2-edge connected. Intuitively, every edge of the dual graph corresponds to a pair of adjacent cells that are both incident to the same vertex. Aichholzer et al. recently conjectured that given an even number of line-segment obstacles, one can construct a convex partition by successively extending the segments along their supporting lines such that the dual graph is the union of two edge-disjoint spanning trees. Here we present a counterexample to this conjecture, which consists of 16 disjoint line segments, such that the dual graph of any convex partition constructed by this method has a bridge edge, and thus the dual graph cannot be partitioned into two spanning trees. Counterexamples of arbitrarily larger sizes can be constructed similarly. Questions about the dual graph of a convex partition are motivated by the still unresolved conjecture about disjoint compatible geometric matchings by Aichholzer et al.. It has application in the design of fault-tolerant wireless networks in the presence of obstacles (e.g. tall buildings in a city).

A data structure is a repository of information; the goal is to organize the data so that it needs less storage (space) and so that a request for information (query) can be processed quickly. A geometric data structure handles data which have locations attached (e.g. addresses of fire stations in the state of Massachusetts). Geometric data structures have become a pervasive and integral part of life, and can be queried to produce driving directions or the name of the nearest Italian restaurant. Since the space and query time of a data structure depend upon the type of queries it needs to support, it is important to study which tools and techniques are suitable for which data structures. The ongoing quest for better data structures sometimes results in improved methods and sometimes results in entirely new techniques. The goal is to determine optimal data structures with the best possible performance. Given a set S of n points in Rd, a data structure for geometric range searching may report: whether the query range contains any point (emptiness), the number of points in the range (counting), all points in the range (reporting), or the minimum/maximum point

Given a set P of points and a set S of pair-wise disjoint axis-parallel line segments in the plane, we construct a straight line spanning tree T on P such thatevery segment in S crosses at most three edges of T.