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This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k). [ABSTRACT FROM AUTHOR]

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area. [ABSTRACT FROM AUTHOR]

We study the Hausdorff Voronoi diagram of a set 5 of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of 5 under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conquer algorithm for the construction of this diagram that improves upon previous results. As a byproduct we introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI Layout, a measure reflecting the sensitivity of a VLSI design to random manufacturing defects, described in a companion paper. [ABSTRACT FROM AUTHOR]

It is an open problem to find a good characterization for independence or, more generally, the rank function in the d-dimensional rigidity matroid of a graph when d ≥ 3. In this paper we give a brief survey of existing lower and upper bounds on the rank of the 3-dimensional rigidity matroid of a graph and introduce a new upper bound, which may lead to the desired good characterization. [ABSTRACT FROM AUTHOR]

This paper investigates geometric and algorithmic properties of the Voronoi diagram for a transportation network on the Euclidean plane. In the presence of a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm2 + m3 + nm log n) algorithm to compute the Voronoi diagram for a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm + m2 + n log n) time with O(m(n + m)) space and the other O(nm log n + m2log m) time with O(n + m) space. [ABSTRACT FROM AUTHOR]

The 3-D static leaf sequencing (SLS) problem arises in radiation therapy for cancer treatments, aiming to deliver a prescribed radiation dose to a target tumor accurately and efficiently. The treatment time and machine delivery error are two crucial factors to the solution (i.e., a treatment plan) for the SLS problem. In this paper, we prove that the 3-D SLS problem is NP-hard, and present the first ever algorithm for the 3-D SLS problem that can determine a tradeoff between the treatment time and machine delivery error (also called the "tongue-and-groove" error in medical literature). Our new 3-D SLS algorithm with error control gives the users (e.g., physicians) the option of specifying a machine delivery error bound, and subject to the given error bound, the algorithm computes a treatment plan with the minimum treatment time. We formulate the SLS problem with error control as computing a k-weight shortest path in a directed graph and build the graph by computing g-matchings and minimum cost flows. Further, we extend our 3-D SLS algorithm to all the popular radiotherapy machine models with different constraints. In our extensions, we model the SLS problems for some of the radiotherapy systems as computing a minimum g-path cover of a directed acyclic graph. We implemented our new 3-D SLS algorithm suite and conducted an extensive comparison study with commercial planning systems and well-known algorithms in medical literature. Some of our experimental results based on real medical data are presented. [ABSTRACT FROM AUTHOR]

A drawing of a plane graph is called an inner rectangular drawing if every edge is drawn as a horizontal or vertical line segment so that every inner face is a rectangle. In this paper we show that a plane graph G has an inner rectangular drawing D if and only if a new bipartite graph constructed from G has a perfect matching. We also show that D can be found in time O(n1.5/ log n) if G has n vertices and a sketch of the outer face is prescribed, that is, all the convex outer vertices and concave ones are prescribed. [ABSTRACT FROM AUTHOR]

Classical topology-based operations such as extrusion (or more generally Minkowski sums) are defined by a cartesian product of combinatorial structures describing the topology of geometric space subdivisions. In this paper, we define the cartesian product operation for four different structures: semi-simplicial sets, generalized maps, oriented maps and chains of maps. We follow a homogeneous framework allowing to define the cartesian product for any simplicial and cellular structures derived from simplicial sets and combinatorial maps. This results from the relationships already established between these structures and those used in computational geometry and geometric modeling. For esch structure we have studied, we additionally provide a time-optimal computation algorithm. [ABSTRACT FROM AUTHOR]

A two-dimensional direction-length framework is a pair (G, p), where G = (V; D, L) is a graph whose edges are labeled as 'direction' or 'length' edges, and a map p from V to ℝ2. The label of an edge uv represents a direction or length constraint between p(u) and p(v). The framework (G, p) is called globally rigid if every other framework (G, q) in which the direction or length between the endvertices of corresponding edges is the same, is 'congruent' to (G, p), i.e. it can be obtained from (G, p) by a translation and, possibly, a dilation by -1. We show that labeled versions of the two Henneberg operations (0-extension and 1-extension) preserve global rigidity of generic direction-length frameworks. These results, together with appropriate inductive constructions, can be used to verify global rigidity of special families of generic direction-length frameworks. [ABSTRACT FROM AUTHOR]

We prove that computing a geometric minimum-dilation graph on a given set of points in the plane, using not more than a given number of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. We also show that the problem remains NP-hard even when a minimum-dilation tour or path is sought; not even an FPTAS exists in this case. [ABSTRACT FROM AUTHOR]