Publication

Drawing planar graphs of bounded degree with few slopes

János Pach, Balázs Keszegh, Doemoetoer Palvoelgyi
2011
Conference Papers

Abstract

We settle a problem of Dujmovic, Eppstein, Suderman, and Wood by showing that there exists a function f with the property that every planar graph G with maximum degree d admits a drawing with noncrossing straight-line edges, using at most f(d) different slopes. If we allow the edges to be represented by polygonal paths with one bend. then 2d slopes suffice. Allowing two bends per edge, every planar graph with maximum degree d >= 3 can be drawn using segments of at most [d/2] different slopes. There is only one exception: the graph formed by the edges of an octahedron is 4-regular, yet it requires 3 slopes. These bounds cannot be improved.

Official source

https://infoscience.epfl.ch/entities/publication/c78cb18e-747d-4295-b0f2-8d81e8059630

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János Pach, Balázs Keszegh, Doemoetoer Palvoelgyi
2011
Conference Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/c78cb18e-747d-4295-b0f2-8d81e8059630

About this result

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (28)

Planar graph

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.

Dual graph

In the mathematical discipline of graph theory, the dual graph of a planar 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.

Outerplanar graph

In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K4 and K2,3, or by their Colin de Verdière graph invariants. They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2.

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