Table of Contents
What is a planar embedding?
A planar embedding, also called a “plane graph” (Harary 1994, p. 103; Harborth and Möller 1994), “planar drawing,” or “plane drawing,” of a planar graph is an embedding in which no two edges intersect (or overlap) and no two vertices coincide.
Where can I find planar embed?
A graph G= (V, E) is said to be planar if it can be drawn in the plane so that no two edges of G intersect at a point other than a vertex. Such a drawing of a planar graph is called a planar embedding of the graph. For example, K4 is planar since it has a planar embedding as shown in figure 1.8. 1.
Is it possible to embed any plane graph on the surface of a sphere?
Theorem 6.4 A graph can be embedded on the surface of a sphere if it can be embedded in a plane. Proof Consider the stereographic projection of a sphere on the plane. Put the sphere on the plane and call the point of contact as SP (south-pole).
Are planar graphs connected?
Every maximal planar graph is a least 3-connected. If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces.
Are bipartite graphs planar?
Every planar graph whose faces all have even length is bipartite. Special cases of this are grid graphs and squaregraphs, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors. The complete bipartite graph on m and n vertices, denoted by Kn,m is the bipartite graph.
Do planar graphs have to be connected?
Can planar graphs have loops?
Given a connected planar graph G, we construct dual graph G* in three stages. Take a plane drawing of G. Choose one point inside each face of the plane drawing – these points are the vertices of G*. For each e of the plane drawing, draw a line connecting the vertices of G* on each side of e.
Where are planar graphs used?
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….Planar graph.
| Example graphs | |
|---|---|
| Planar | Nonplanar |
| Butterfly graph | Complete graph K5 |
| Complete graph K4 | Utility graph K3,3 |