Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. Now imagine the set P is partitioned into two disjoint subsets R and B, and we ask for an embedding of T in P without crossings and with the property that all edges join a point in R (red) and a point in B (blue). In this case we say that T admits a bipartite embedding with respect to the bipartition (R, B). Examples show that the problem in its full generality is not solvable. In view of this fact we consider several embedding problems and study for which bipartitions they can be solved. We present several results that are valid for any bipartition (R, B) in general position, and some other results that hold for particular configurations of points.
In this article, we focus on structural properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus (undirected) trees. For some of these properties, we give the matrix version, regarding nearly reducible matrices. We give bounds for the coefficients of the characteristic polynomial corresponding to the adjacency matrix of trees, and we conjecture bounds for MSDs. We also propose two different representations of an MSD in terms of trees (the union of a spanning tree and a directed forest; and a double directed tree whose vertices are given by the contraction of connected Hasse diagrams).
We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We characterize the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove that every minimal strong digraph of order n ≥ 2 is the expansion of a minimal strong digraph of order n − 1 and we give sequentially generative procedures for the constructive characterization of the classes of minimal strong digraphs. Finally we describe algorithms to compute unlabeled minimal strong digraphs and their isospectral classes.2010-AMS Classification: 05C20, 05C40, 05C75.
Given a set of points S = {pi,...,pn} in the euclidean d-dimensional space, we address the problem of computing the d-dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d = 2, a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace angle-wise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm: it involves the computation of furthest and closest point Voronoi diagrams and a pruneand-search in the former one.
We study the feasibility of design for a layer-deposition manufacturing process called stereolithography which w orks by controlling a vertical laser beam which when targeted on a photocurable liquid causes the liquid to harden. In order to better understand the power as well as the limitations of this manufacturing process, we de ne a mathematical model of stereolithography referred to as vertical stereolithography and analyze the class of objects that can be constructed under the assumptions of the model. Given an object modelled as a polygon or a polyhedron, we give algorithms that decide in On time whether or not the object can be constructed by v ertical stereolithography. If the answer is in the a rmative, the algorithm reports a description of all the orientations in which the object can be made. We then de ne a more exible model that more accurately re ects the actual capabilities of stereolithography referred to as variable-angle stereolithography and again study the class of feasible objects for this model. We give a n On time algorithm for polygons and On logn a s w ell as On time algorithms for polyhedra. We show that objects formed with variable-angle stereolithography can also be constructed using another manufacturing process known as gravity casting. F urthermore, we show that the polyhedral objects formed by v ertical stereolithography are closely related to polyhedral terrains which are important structures in geographic information systems GIS and computational geometry. In fact, an object built with variable-angle stereolithography resembles a terrain with overhangs, thus initiating the study of more realistic terrains than the standard ones considered in geographic information systems. Finally, w e relate our results to the area of grasping in robotics by showing that the polygonal and polyhedral objects that can be built by v ertical stereolithography can be clamped by parallel jaw grippers with any positive sized gripper.
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