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We determine the reachability properties of the embeddings in R3 of a directed path, in the graph-theoretic sense, whose edges have each been assigned a desired direction (East, West, North, South, Up, or Down) but no length. We ask which points of R3 can be reached by the terminus of an embedding of such a path, by choosing appropriate positive lengths for the edges, if the embedded path starts at the origin, does not intersect itself, and respects the directions pre-assigned to its edges. This problem arises in the context of extending planar graph embedding techniques and VLSI rectilinear layout techniques from 2D to 3D. We give a combinatorial characterization of reachability that yields linear time recognition and layout algorithms. Finally, we extend our characterization to Rd, d>3. [Copyright &y& Elsevier]

We establish new results on the curve complexity of k -colored point-set embeddings when k = 3. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require Ω (n 1 3 ) bends on Ω (n 2 3 ) edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embeddings of k -colored trees and it extends a lower bound by Pach and Wenger [35] to the case that the graph only has O (1) independent edges. Concerning upper bounds, we prove that any 3-colored path admits a 3-colored point-set embedding with curve complexity at most 4. In addition, we introduce a variant of the k -colored simultaneous embeddability problem and study its relationship with the k -colored point-set embeddability problem. [ABSTRACT FROM AUTHOR]

Abstract: The contribution of this paper is twofold. It presents a new approach to the matched drawability problem of pairs of planar graphs and it provides four algorithms based on this approach for drawing the pairs , , and . Further, it initiates the study of the matched drawability of triples of planar graphs: it presents an algorithm to compute a matched drawing of a triple of cycles and an algorithm to compute a matched drawing of a caterpillar and two unlabeled level planar graphs. The results extend previous work on the subject and relate to existing literature about simultaneous embeddability and unlabeled level planarity. [Copyright &y& Elsevier]

Abstract: Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of . We concentrate on trees and show how to compute the output in time in a real-RAM model and with at most edges with at most bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least bends on edges. [Copyright &y& Elsevier]

Abstract: Let be a planar graph with vertices and with a partition of the vertex set into subsets for some positive integer . Let be a set of distinct points in the plane with a partition into subsets with (). This paper studies the problem of computing a planar polyline drawing of , such that each vertex of is mapped to a distinct point of . Lower and upper bounds on the number of bends per edge are proved for any . In the special case , we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717–728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest. [Copyright &y& Elsevier]

Abstract: In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position. [Copyright &y& Elsevier]