612 research outputs found

    Strongly Monotone Drawings of Planar Graphs

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    A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices. We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs are based on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex

    Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

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    A greedily routable region (GRR) is a closed subset of R2\mathbb R^2, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201

    Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

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    A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is xx- and yy-monotone. Angle-monotone graphs are 2\sqrt 2-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-Ξ6\theta_6-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex ss to any vertex tt whose length is within 1+21 + \sqrt 2 times the Euclidean distance from ss to tt. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Drawing Graphs as Spanners

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    We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph GG, the goal is to construct a straight-line drawing Γ\Gamma of GG in the plane such that, for any two vertices uu and vv of GG, the ratio between the minimum length of any path from uu to vv and the Euclidean distance between uu and vv is small. The maximum such ratio, over all pairs of vertices of GG, is the spanning ratio of Γ\Gamma. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 11, a proper straight-line drawing with spanning ratio 11, and a planar straight-line drawing with spanning ratio 11 are NP-complete, ∃R\exists \mathbb R-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio 11 to spanning ratio 1+Ï”1+\epsilon allows us to draw every graph. Namely, we prove that, for every Ï”>0\epsilon>0, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than 1+Ï”1+\epsilon. Third, our drawings with spanning ratio smaller than 1+Ï”1+\epsilon have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio

    On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs

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    In this paper we study the area requirements of planar greedy drawings of triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family H\cal H of subdivisions of triconnected plane graphs and claimed that every planar greedy drawing of the graphs in H\mathcal H respecting the prescribed plane embedding requires exponential area. However, we show that every nn-vertex graph in H\cal H actually has a planar greedy drawing respecting the prescribed plane embedding on an O(n)×O(n)O(n)\times O(n) grid. This reopens the question whether triconnected planar graphs admit planar greedy drawings on a polynomial-size grid. Further, we provide evidence for a positive answer to the above question by proving that every nn-vertex Halin graph admits a planar greedy drawing on an O(n)×O(n)O(n)\times O(n) grid. Both such results are obtained by actually constructing drawings that are convex and angle-monotone. Finally, we consider α\alpha-Schnyder drawings, which are angle-monotone and hence greedy if α≀30∘\alpha\leq 30^\circ, and show that there exist planar triangulations for which every α\alpha-Schnyder drawing with a fixed α<60∘\alpha<60^\circ requires exponential area for any resolution rule

    On Planar Greedy Drawings of 3-Connected Planar Graphs

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    A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra\u27s result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture
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