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In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree. [ABSTRACT FROM AUTHOR]

There is strong empirical evidence that human perception of a graph drawing is negatively correlated with the number of edge crossings. However, recent experiments show that one can reduce the negative effect by ensuring that the edges that cross do so at large angles. These experiments have motivated a number of mathematical and algorithmic studies of 'right angle crossing (RAC)' drawings of graphs, where the edges cross each other perpendicularly. In this paper we give an algorithm for constructing RAC drawings of 'outer-1-plane' graphs, that is, topological graphs in which each vertex appears on the outer face, and each edge crosses at most one other edge. The drawing algorithm preserves the embedding of the input graph. This is one of the few algorithms available to construct RAC drawings. [ABSTRACT FROM AUTHOR]

It has been shown that there exist planar digraphs that require exponential area in every upward straight-line planar drawing. On the other hand, upward poly-line planar drawings of planar graphs can be realized in Θ(n2) area. In this paper we consider families of DAGs that naturally arise in practice, like DAGs whose underlying graph is a tree (directed trees), is a bipartite graph (directed bipartite graphs), or is an outerplanar graph (directed outerplanar graphs). Concerning directed trees, we show that optimal Θ(n log n) area upward straight-line/poly-line planar drawings can be constructed. However, we prove that if the order of the neighbors of each node is assigned, then exponential area is required for straight-line upward drawings and quadratic area is required for poly-line upward drawings, results surprisingly and sharply contrasting with the area bounds for planar upward drawings of undirected trees. After having established tight bounds on the area requirements of planar upward drawings of several families of directed trees, we show how the results obtained for trees can be exploited to determine asymptotic optimal values for the area occupation of planar upward drawings of directed bipartite graphs and directed outerplanar graphs. [ABSTRACT FROM AUTHOR]

Let G1 and G2 be two planar graphs having some vertices in common. A simultaneous embedding of G1 and G2 is a pair of crossing-free drawings of G1 and G2 such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle. [ABSTRACT FROM AUTHOR]