Straight-line Embedding Research Articles

A Logarithmic Bound for Simultaneous Embeddings of Planar Graphs

AbstractA set $${\mathcal {G}}$$ G of planar graphs on the same number n of vertices is called simultaneously embeddable if there exists a set P of n points in the plane such that every graph $$G \in {\mathcal {G}}$$ G ∈ G admits a (crossing-free) straight-line embedding with vertices placed at points of P. A conflict collection is a set of planar graphs of the same order with no simultaneous embedding. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether there exists a conflict collection of size 2. While this remains widely open, we give a short proof that for sufficiently large n there exists a conflict collection consisting of at most $$(3+o(1))\log _2(n)$$ ( 3 + o ( 1 ) ) log 2 ( n ) planar graphs on n vertices. This constitutes a double-exponential improvement over the previously best known bound of $$O(n\cdot 4^{n/11})$$ O ( n · 4 n / 11 ) for the same problem by Goenka et al. (Graphs Combin 39:100, 2023). Using our method we also provide a computer-free proof that for every integer $$n\in [107,193]$$ n ∈ [ 107 , 193 ] there exists a conflict collection of 30 planar n-vertex graphs, improving upon the previously smallest known conflict collection consisting of 49 graphs of order 11, which was found using heavy computer assistance. While the construction by Goenka et al. was explicit, our construction of a conflict collection of size $$O(\log n)$$ O ( log n ) is based on the probabilistic method and is thus only implicit. Motivated by this, for every large enough n we give a different, fully explicit construction of a collection of less than $$n^6$$ n 6 planar n-vertex graphs with no simultaneous embedding.

How to Fit a Tree in a Box

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with n nodes can be drawn on a \(\sqrt{n}\) by \(\sqrt{n}\) grid. We also show that testing whether a given rooted binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.

PTOPO: Computing the geometry and the topology of parametric curves

We consider the problem of computing the topology and describing the geometry of a parametric curve in Rn. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization.Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most d and maximum bitsize of coefficients τ, then the worst case bit complexity of PTOPO is O˜B(nd6+nd5τ+d4(n2+nτ)+d3(n2τ+n3)+n3d2τ). This bound matches the current record bound O˜B(d6+d5τ) for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if N=max⁡{d,τ}, the complexity of PTOPO becomes O˜B(N6), which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of N10. In the same time complexity, we obtain a graph whose straight-line embedding is isotopic to the curve. However, visualizing the curve on top of the abstract graph construction, increases the bound to O˜B(N7). For curves of general dimension, we can also distinguish between ordinary and non-ordinary real singularities and determine their multiplicities in the same expected complexity of PTOPO by employing the algorithm of Blasco and Pérez-Díaz [CAGD'19]. We have implemented PTOPO in maple for the case of plane and space curves. Our experiments illustrate its practical nature.

Grid straight-line embeddings of trees with a minimum number of bends per path

A grid embedding of a tree T whose vertices have degree at most four is an embedding in which the vertices of T are represented by points with integer coordinates, and the edges are represented by interior-disjoint horizontal or vertical line segments joining pairs of adjacent vertices in T. In any such embedding, a path of T may bend several times. In this paper we obtain a linear-time algorithm that given a tree T finds a grid embedding of T that minimizes the maximum number of times any path of T bends. All of the results presented here generalize easily to lattice embeddings of trees in higher dimensions.

A Note on Universal Point Sets for Planar Graphs

We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that at least $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex planar graph (vertices are drawn as the given points); this improves the previous best constant $1.235$ by Kurowski (2004). Our second main result is based on exhaustive computer search: We show that no set of 11 points exists, on which all planar 11-vertex graphs can be simultaneously drawn plane straight-line. This strengthens the result by Cardinal, Hoffmann, and Kusters (2015), that all planar graphs on $n \le 10$ vertices can be simultaneously drawn on particular $n$-universal sets of $n$ points while there are no $n$-universal sets of size $n$ for $n \ge 15$. We also provide 49 planar 11-vertex graphs which cannot be simultaneously drawn on any set of 11 points. This, in fact, is another step towards a (negative) answer of the question, whether every two planar graphs can be drawn simultaneously - a question raised by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw, and Mitchell (2007).

Embedding of fuzzy graphs on topological surfaces

Planar graph is a special type in crisp as well as in fuzzy graphs. In fuzzy planar graphs, the planarity value is the amount of planarity of the crossed fuzzy edges, so that the intersection of fuzzy edges are possible in fuzzy graphs as compared to the planar graphs in crisp. Generally, the fuzzy planar graphs are depicted in the plane surface. In this paper, the embedding of fuzzy graphs are discussed in the surfaces like sphere and m-torus. Moreover, definition of fuzzy planar triangulation, straight-line, and piecewise embedding are also stated for planar embedding. Some of the effective definitions and theorems are illustrated with examples. Theorems like Euler’s formula for plane and sphere surfaces are proved and formulated for fuzzy planar graphs.

Small Universal Point Sets for k-Outerplanar Graphs

A point set $$\mathcal{S} \subseteq \mathbb {R}^2$$ is universal for a class $$\mathcal G$$ of planar graphs if every graph of $$\mathcal{G}$$ has a planar straight-line embedding on $$\mathcal{S}$$ . It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an $$O(n \log n)$$ -size universal point set in two distinct cases, namely when $$k=2$$ (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).

Recurrence of multiply-ended planar triangulations

In this note we show that a bounded degree planar triangulation is recurrent if and only if the set of accumulation points of some/any circle packing of it is polar (that is, planar Brownian motion avoids it with probability 1). This generalizes a theorem of He and Schramm [6] who proved it when the set of accumulation points is either empty or a Jordan curve, in which case the graph has one end. We also show that this statement holds for any straight-line embedding with angles uniformly bounded away from 0.

Free Edge Lengths in Plane Graphs

We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if all constraints on the edge lengths of G depend on G only, irrespective of additional edges of the host H. We characterize the planar graphs G that are free in every host H, $$G\subseteq H$$G⊆H, and all the planar graphs G that are extrinsically free in every host H, $$G\subseteq H$$G⊆H. The case of cycles $$G=C_k$$G=Ck provides a new version of the celebrated carpenter's rule problem. Even though cycles $$C_k$$Ck, $$k\ge 4$$k?4, are not extrinsically free in all triangulations, it turns out that nondegenerate edge lengths are always realizable, where the edge lengths are considered degenerate if the cycle can be flattened (into a line) in two different ways. Separating triangles, and separating cycles in general, play an important role in our arguments. We show that every star is free in a 4-connected triangulation (which has no separating triangle).

Universal point sets for planar three-trees

For every n∈N, we present a set Sn of O(n3/2log⁡n) points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of Sn. This is the first subquadratic upper bound on the cardinality of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.

On Universal Point Sets for Planar Graphs

A set P of points in R2 is n-universal if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n ≥ 15. Conversely, we use a computer program to show that there exist universal point sets for all n ≤ 10 and to enumerate all corresponding order types. Finally, we describe a collection G of 7′393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in G.

Embedding the dual complex of hyper-rectangular partitions

A rectangular partition is the partition of an (axis-aligned) rectangle into interior-disjoint rectangles. We ask whether a rectangular partition permits a nice drawing of its dual, that is, a straight-line embedding of it such that each dual vertex is placed into the rectangle that it represents. We show that deciding whether such a drawing exists is NP-complete. Moreover, we consider the drawing where a vertex is placed in the center of the representing rectangle and consider sufficient conditions for this drawing to be nice. This question is studied both in the plane and for the higher-dimensional generalization of rectangular partitions.

Euclidean chains and their shortcuts

A Euclidean graph is a straight line embedding of a graph in the plane such that there is no crossing between any pair of edges and the length of an edge is the Euclidean distance between its two endpoints. A Euclidean chain C=(x1,x2,…,xn) is a planar straight line graph with vertex set {x1,x2,…,xn} and edge set {xixi+1:1≤i≤n−1}. Given a Euclidean chain C in the plane, we study the problem of finding a pair of points on C such that the new Euclidean graph obtained from C by adding a straight line segment (called a shortcut) between this pair of points has the minimum diameter. We also study the ratio between the diameter of the new graph and the length of C. We give necessary and sufficient conditions for optimal shortcuts. We present an approximation algorithm for computing the optimal shortcuts of chains. In addition, we introduce two types of chains: a strongly monotonic chain and a simple chain. We provide properties for these two types of chains and their shortcuts.

Greenʼs theorem and isolation in planar graphs

We show a simple application of Greenʼs theorem from multivariable calculus to the isolation problem in planar graphs. In particular, we give a log-space construction of a skew-symmetric, polynomially-bounded edge weight function for directed planar graphs, such that the weight of any simple cycle in the graph is non-zero with respect to this weight function. As a direct consequence of the above weight function, we are able to isolate a directed path between two fixed vertices, in a directed planar graph. We also show that given a bipartite planar graph, we can obtain an edge weight function (using the above function) in log-space, which isolates a perfect matching in the given graph. Earlier this was known to be true only for grid graphs – which is a proper subclass of planar graphs.We also look at the problem of obtaining a straight line embedding of a planar graph in log-space. Although we do not quite achieve this goal, we give a piecewise straight line embedding of the given planar graph in log-space.

On a Tree and a Path with no Geometric Simultaneous Embedding

Two graphs $G_1=(V,E_1)$ and $G_2=(V,E_2)$ admit a geometric simultaneous embedding if there exists a set of points P and a bijection M: P -> V that induce planar straight-line embeddings both for $G_1$ and for $G_2$. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has depth 4.

Spanners of additively weighted point sets

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs ( p , r ) where p is a point in the plane and r is a real number. The distance between two points ( p i , r i ) and ( p j , r j ) is defined as | p i p j | − r i − r j . We show that in the case where all r i are positive numbers and | p i p j | ⩾ r i + r j for all i, j (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a ( 1 + ϵ ) -spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has a spanning ratio bounded by a constant. The straight-line embedding of the Additively Weighted Delaunay graph may not be a plane graph. Given the Additively Weighted Delaunay graph, we show how to compute a plane straight-line embedding that also has a spanning ratio bounded by a constant in O ( n log n ) time.

Upward straight-line embeddings of directed graphs into point sets

In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position.

Connectivity check in 3-connected planar graphs with obstacles

Abstract We define a vertex labelling for every planar 3-connected graph with n vertices from which one can answer connectivity queries. A connectivity query asks whether there exists in the given graph a path linking u and v that avoids a set F of edges and a set X of vertices. The vertices u,v and those of X are given by their labels. The edges of F are given by the labels of their ends. Each label has a size of O ( log ( n ) ) bits. Our construction makes an essential use of straight-line embeddings on n × n grids of simple loop-free planar graphs. Such embeddings can be constructed in linear time by Schnyder's algorithm [W. Schnyder. Embedding Planar Graphs in the Grid. SODA, First ACM-SIAM Symposium on Discrete Algorithms, San Francisco, pages 138–148, 1990] (see also [H. de Fraysseix, J. Pach, and R. Pollack. Small Sets Supporting Fary Embeddings of Planar Graphs. In Twentieth Annual ACM Symposium on Theory of Computing, pages 426–433. ACM, 1988; H. de Fraysseix, J. Pach, and R. Pollack. How to Draw a Planar Graph on a Grid. Combinatorica, 10:41–51, 1990]).

Planar Straight-Line Embedding of Double-Tree Scan Architecture on a Rectangular Grid

Double-tree-scan (DTS) is a new scan-path architecture that is deemed to be suitable for low-power testing of VLSI circuits. A full DTS resembles two complete k-level (k > 0) binary trees whose leaf nodes are merged pair-wise, and thus consists of exactly N$_{k}$ = 3 × 2$^{k}$ − 2 nodes. In this paper, the problem of planar straight-line embedding of a graph on a rectangular grid is investigated and an O(N$_{k}$) time algorithm for drawing it, is described. The embedding requires at most 2N$_{k}$ grid points, with an aspect ratio lying between 1 and f. Next, techniques of embedding a partial DTS is considered when the number of nodes n ≠ 3 × 2$^{k}$ − 2, for some k. Layouts of double-tree scan-paths for some benchmark circuits are also presented to demonstrate the results.

Plane Embeddings of Planar Graph Metrics

Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires \(\Theta(\sqrt n)\) distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by Matousek [M3] by proving that some planar graph metrics require \(\Omega(n^{2/3})\) distortion in any embedding into the plane, proving the first separation between these two types of graph metrics. We also prove that some planar graph metrics require \(\Omega(n)\) distortion in any crossing-free straight-line embedding into the plane, suggesting a separation between low-distortion plane embedding and the well-studied notion of crossing-free straight-line planar drawings. Finally, on the upper-bound side, we prove that all outerplanar graph metrics can be embedded into the plane with \(O(\sqrt n)\) distortion, generalizing the previous results on trees (both the worst-case bound and the approximation algorithm) and building techniques for handling cycles in plane embeddings of graph metrics.