126,697 research outputs found

    The Complexity of Drawing a Graph in a Polygonal Region

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    We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This strengthens an NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to be bounded in the special case of a convex region, or for drawing a path in any polygonal region.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

    The Complexity of Drawing a Graph in a Polygonal Region

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    We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This establishes a wider context for the NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to have bounded bit complexity for the special case of a convex region, or for drawing a path in any polygonal region. In addition, we prove a Mnëv-type universality result—loosely speaking, that the solution spaces of instances of our graph drawing problem are equivalent, in a topological and algebraic sense, to bounded algebraic varieties

    β\beta-Stars or On Extending a Drawing of a Connected Subgraph

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    We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing ΓH\Gamma_H of an induced connected subgraph HH can be extended with at most min{h/2,β+log2(h)+1}\min\{ h/2, \beta + \log_2(h) + 1\} bends per edge, where β\beta is the largest star complexity of a face of ΓH\Gamma_H and hh is the size of the largest face of HH. This result significantly improves the previously known upper bound of 72V(H)72|V(H)| [5] for the case where HH is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

    Planar Drawings of Fixed-Mobile Bigraphs

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    A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

    Drawing Planar Graphs with a Prescribed Inner Face

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    Given a plane graph GG (i.e., a planar graph with a fixed planar embedding) and a simple cycle CC in GG whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of GG. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time

    The Partial Visibility Representation Extension Problem

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    For a graph GG, a function ψ\psi is called a \emph{bar visibility representation} of GG when for each vertex vV(G)v \in V(G), ψ(v)\psi(v) is a horizontal line segment (\emph{bar}) and uvE(G)uv \in E(G) iff there is an unobstructed, vertical, ε\varepsilon-wide line of sight between ψ(u)\psi(u) and ψ(v)\psi(v). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph GG, a bar visibility representation ψ\psi of GG, additionally, puts the bar ψ(u)\psi(u) strictly below the bar ψ(v)\psi(v) for each directed edge (u,v)(u,v) of GG. We study a generalization of the recognition problem where a function ψ\psi' defined on a subset VV' of V(G)V(G) is given and the question is whether there is a bar visibility representation ψ\psi of GG with ψ(v)=ψ(v)\psi(v) = \psi'(v) for every vVv \in V'. We show that for undirected graphs this problem together with closely related problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Drawing Trees with Perfect Angular Resolution and Polynomial Area

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    We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure
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