Planar embeddability of the vertices of a graph using a fixed point set is NP-hard

Let G = (V, E) be a graph with n vertices and let P be a set of n points in the plane. We show that deciding whether there is a planar straight-line embedding of G such that the vertices V are embedded onto the points P is NP-complete, even when G is 2-connected and 2-outerplanar. This settles an open problem posed in [P. Bose. On embedding an outer-planar graph in a point set. Comput. Geom. Theory Appl., 23:303-312, November 2002. A preliminary version appeared in Graph Drawing (Proc. GD ’97), LNCS 1353, pg. 25-36, F. Brandenberg, D. Eppstein, M.T. Goodrich, S.G. Kobourov, G. Liotta, and P. Mutzel. Selected open problems in graph drawing. In Graph Drawing (Proc. GD’03), LNCS, 2003. To appear y M. Kaufmann and R. Wiese. Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications, 6(1):115–129, 2002. A preliminary version appeared in Graph Drawing (Proc. GD ’99), LNCS 1731, pg. 165–174].Cornelis Lely Stichtin

Simultaneous Embeddability of Two Partitions

We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201

On Embeddability of Buses in Point Sets

Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the \emph{bus embeddability problem} (BEP): given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT algorithm for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201

Upward Point-Set Embeddability

We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph $D$ has an upward planar embedding into a point set $S$. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of $k$-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a $1$-switch tree), we show that not every $k$-switch tree admits an upward planar straight-line embedding into any convex point set, for any $k \geq 2$. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete

Improved Algorithms for the Point-Set Embeddability problem for Plane 3-Trees

In the point set embeddability problem, we are given a plane graph $G$ with $n$ vertices and a point set $S$ with $n$ points. Now the goal is to answer the question whether there exists a straight-line drawing of $G$ such that each vertex is represented as a distinct point of $S$ as well as to provide an embedding if one does exist. Recently, in \cite{DBLP:conf/gd/NishatMR10}, a complete characterization for this problem on a special class of graphs known as the plane 3-trees was presented along with an efficient algorithm to solve the problem. In this paper, we use the same characterization to devise an improved algorithm for the same problem. Much of the efficiency we achieve comes from clever uses of the triangular range search technique. We also study a generalized version of the problem and present improved algorithms for this version of the problem as well