Convex drawing

Last updated
Convex and strictly convex grid drawings of the same graph Convex and strictly convex grid drawings.svg
Convex and strictly convex grid drawings of the same graph

In graph drawing, a convex drawing of a planar graph is a drawing that represents the vertices of the graph as points in the Euclidean plane and the edges as straight line segments, in such a way that all of the faces of the drawing (including the outer face) have a convex boundary. The boundary of a face may pass straight through one of the vertices of the graph without turning; a strictly convex drawing asks in addition that the face boundary turns at each vertex. That is, in a strictly convex drawing, each vertex of the graph is also a vertex of each convex polygon describing the shape of each incident face.

Every polyhedral graph has a strictly convex drawing, [1] for instance obtained as the Schlegel diagram of a convex polyhedron representing the graph. For these graphs, a convex (but not necessarily strictly convex) drawing can be found within a grid whose length on each side is linear in the number of vertices of the graph, in linear time. [2] [3] However, strictly convex drawings may require larger grids; for instance, for any polyhedron such as a pyramid in which one face has a linear number of vertices, a strictly convex drawing of its graph requires a grid of cubic area. [4] A linear-time algorithm can find strictly convex drawings of polyhedral graphs in a grid whose length on each side is quadratic. [5]

Convex but not strictly convex drawing of the complete bipartite graph K 2 , 3 {\displaystyle K_{2,3}} Graph homeomorphism example 1.svg
Convex but not strictly convex drawing of the complete bipartite graph

Other graphs that are not polyhedral can also have convex drawings, or strictly convex drawings. Some graphs, such as the complete bipartite graph , have convex drawings but not strictly convex drawings. A combinatorial characterization for the graphs with convex drawings is known, [6] and they can be recognized in linear time, [7] but the grid dimensions needed for their drawings and an efficient algorithm for constructing small convex grid drawings of these graphs are not known in all cases. [8]

Convex drawings should be distinguished from convex embeddings, in which each vertex is required to lie within the convex hull of its neighboring vertices. Convex embeddings can exist in dimensions other than two, do not require their graph to be planar, and even for planar embeddings of planar graphs do not necessarily force the outer face to be convex. [9]

Related Research Articles

Dual polyhedron Polyhedron associated with another by swapping vertices for faces

In geometry, every polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Convex polytope Convex hull of a finite set of points in a Euclidean space

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

Dual graph Graph representing faces of another graph

In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.

Geometric graph theory Subfield of graph theory

Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs". Geometric graphs are also known as spatial networks.

In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently by Klaus Wagner (1936), Fáry (1948), and Sherman K. Stein (1951).

Graph embedding Embedding a graph in a topological space, often Euclidean

In topological graph theory, an embedding of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs are associated with edges in such a way that:

Halin graph Mathematical tree with cycle through leaves

In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of its edges cross, and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.

Polyhedral graph Graph made from vertices and edges of a convex polyhedron

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs.

Herschel graph

In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph, and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs.

Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.

Apex graph Graph which can be made planar by removing a single node

In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K5 or K3,3, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.

Apollonian network Graph formed by subdivision of triangles

In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.

1-planar graph

In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. If a 1-planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1-plane graph or 1-planar embedding of the graph.

In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by W. T. Tutte (1963), states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.

In distributed computing and geometric graph theory, greedy embedding is a process of assigning coordinates to the nodes of a telecommunications network in order to allow greedy geographic routing to be used to route messages within the network. Although greedy embedding has been proposed for use in wireless sensor networks, in which the nodes already have positions in physical space, these existing positions may differ from the positions given to them by greedy embedding, which may in some cases be points in a virtual space of a higher dimension, or in a non-Euclidean geometry. In this sense, greedy embedding may be viewed as a form of graph drawing, in which an abstract graph is embedded into a geometric space.

Upward planar drawing Graph with edges non-crossing and upward

In graph drawing, an upward planar drawing of a directed acyclic graph is an embedding of the graph into the Euclidean plane, in which the edges are represented as non-crossing monotonic upwards curves. That is, the curve representing each edge should have the property that every horizontal line intersects it in at most one point, and no two edges may intersect except at a shared endpoint. In this sense, it is the ideal case for layered graph drawing, a style of graph drawing in which edges are monotonic curves that may cross, but in which crossings are to be minimized.

In graph drawing, the area used by a drawing is a commonly used way of measuring its quality.

References

  1. ↑ Tutte, W. T. (1960), "Convex representations of graphs", Proceedings of the London Mathematical Society, Third Series, 10: 304–320, doi:10.1112/plms/s3-10.1.304, MR   0114774
  2. ↑ Kant, G. (1996), "Drawing planar graphs using the canonical ordering", Algorithmica, 16 (1): 4–32, doi:10.1007/s004539900035, hdl: 1874/16676 , MR   1394492
  3. ↑ Bonichon, Nicolas; Felsner, Stefan; Mosbah, Mohamed (2007), "Convex drawings of 3-connected plane graphs", Algorithmica, 47 (4): 399–420, doi:10.1007/s00453-006-0177-6, MR   2304987, S2CID   375595
  4. ↑ Andrews, George E. (1963), "A lower bound for the volume of strictly convex bodies with many boundary lattice points", Transactions of the American Mathematical Society, 106 (2): 270–279, doi: 10.2307/1993769 , JSTOR   1993769, MR   0143105
  5. ↑ Bárány, Imre; Rote, Günter (2006), "Strictly convex drawings of planar graphs", Documenta Mathematica, 11: 369–391, arXiv: cs/0507030 , MR   2262937
  6. ↑ Thomassen, Carsten (1980), "Planarity and duality of finite and infinite graphs", Journal of Combinatorial Theory, Series B, 29 (2): 244–271, doi: 10.1016/0095-8956(80)90083-0 , MR   0586436
  7. ↑ Chiba, Norishige; Yamanouchi, Tadashi; Nishizeki, Takao (1984), "Linear algorithms for convex drawings of planar graphs", in Bondy, J. Adrian; Murty, U. S. R. (eds.), Progress in graph theory (Waterloo, Ont., 1982), Academic Press, pp. 153–173, MR   0776799
  8. ↑ Zhou, Xiao; Nishizeki, Takao (2010), "Convex drawings of internally triconnected plane graphs on grids", Discrete Mathematics, Algorithms and Applications, 2 (3): 347–362, doi:10.1142/S179383091000070X, MR   2728831
  9. ↑ Linial, N.; Lovász, L.; Wigderson, A. (1988), "Rubber bands, convex embeddings and graph connectivity", Combinatorica, 8 (1): 91–102, doi:10.1007/BF02122557, MR   0951998, S2CID   6164458
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.