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In this paper, we show a polynomial-time algorithm for testing c -planarity of embedded flat clustered graphs with at most two vertices per cluster on each face. Our result is based on a reduction to the planar set of spanning trees in topological multigraphs (pssttm) problem, which is defined as follows. Given a (non-planar) topological multigraph A with k connected components A 1 , ... , A k , do spanning trees of A 1 , ... , A k exist such that no two edges in any two spanning trees cross? Kratochvíl et al. [SIAM Journal on Discrete Mathematics, 4(2): 223–244, 1991] proved that the problem is NP-hard even if k = 1 ; on the other hand, Di Battista and Frati presented a linear-time algorithm to solve the pssttm problem for the case in which A is a 1 -planar topological multigraph [Journal of Graph Algorithms and Applications, 13(3): 349–378, 2009]. For any embedded flat clustered graph C , an instance A of the pssttm problem can be constructed in polynomial time such that C is c -planar if and only if A admits a solution. We show that, if C has at most two vertices per cluster on each face, then it can be tested in polynomial time whether the corresponding instance A of the pssttm problem is positive or negative. Our strategy for solving the pssttm problem on A is to repeatedly perform a sequence of tests, which might let us conclude that A is a negative instance, and simplifications, which might let us simplify A by removing or contracting some edges. Most of these tests and simplifications are performed "locally", by looking at the crossings involving a single edge or face of a connected component A i of A ; however, some tests and simplifications have to consider certain global structures in A , which we call α -donuts. If no test concludes that A is a negative instance of the pssttm problem, then the simplifications eventually transform A into an equivalent 1 -planar topological multigraph on which we can apply the cited linear-time algorithm by Di Battista and Frati. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In this paper we consider the problem of testing an arbitrary digraph G = (V,E) for upward planarity. In particular we describe two fixed-parameter tractable algorithms for testing the upward planarity of G. The first algorithm that we present can test the upward planarity of G in O(2t · t! · |V|2)-time where t is the number of triconnected components of G. The second algorithm that we present uses a standard technique known as kernelisation and can test the upward planarity of G in O(|V|2 + k4 · (2k)!)-time, where k = |E| - |V|. [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]

An h-layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end-vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper h-layer drawing contains only proper edges, a short h-layer drawing contains no long edges, an upright h-layer drawing contains no flat edges, and an unconstrained h-layer drawing contains any type of edge. In this paper, we derive upper and lower bounds on the number of layers required by proper, short, upright, and unconstrained layered drawings of trees. We prove that these bounds are optimal with respect to the pathwidth of the tree being drawn. Finally, we give linear-time algorithms for obtaining layered drawings that match these upper bounds. [ABSTRACT FROM AUTHOR]