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bogdan
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What is the definition of a planar graph and which properties does it have ?
Planar Graphs: Definition & Properties
- Thread starter bogdan
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In summary, a planar graph is defined as one that can be drawn in a plane with no self-intersections. It has the property of only having relations between its components, and there are theorems that determine whether a graph is planar or not. These theorems are described in books by F. Harary and O. Ore, which are not available online, but a similar resource can be found on the website provided.
- #2
lethe
- 653
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a planar graph is one which can be drawn in a plane with no self-intersections. i believe the simplest nonplanar graph is the inscribed pentagram.
- #3
bogdan
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Well...I know that...
But isn't there an "abstract" definition which explains what a planar graph is without using the concept of "plan" ?
Only relations between the components of the graph...
But isn't there an "abstract" definition which explains what a planar graph is without using the concept of "plan" ?
Only relations between the components of the graph...
- #4
lethe
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there are some theorems about when a graph is planar or not. but you want to do what now? define a planar graph without using a plane?
- #5
bogdan
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Aaa...those theorems sound interesting...that's what I need...
Where can I find them ?
(I want to define a plane using a planar graph )
Where can I find them ?
(I want to define a plane using a planar graph )
- #6
rutwig
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See the book of F. Harary. He gives a complete review starting from the famous theorem of Kuratowski. The booklet by O. Ore does also deal a bit with planarity of graphs.
- #7
bogdan
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If only I could find those books...anyway...I fount something on the Internet...but it's so complicated...
- #8
rutwig
- 44
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Originally posted by bogdan
If only I could find those books...anyway...I fount something on the Internet...but it's so complicated...
The books cited are not available on the net, since they were written in 1967 and 1968. However, have a look at
http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/download.html
FAQ: Planar Graphs: Definition & Properties
What is a planar graph?
A planar graph is a type of graph in which the edges do not intersect each other when drawn on a plane. This means that the edges can be drawn without crossing over each other, making it a useful tool for representing networks and relationships between objects.
What are the properties of planar graphs?
Some key properties of planar graphs include: having no edge intersections, being able to be drawn on a 2-dimensional plane, having a unique dual graph, and being able to be represented by a set of polygons on the plane.
How are planar graphs useful in real-world applications?
Planar graphs have many practical applications, such as in circuit design, transportation networks, and social network analysis. They can also be used to solve problems in various fields including computer science, biology, and economics.
What is the difference between planar and non-planar graphs?
The main difference between planar and non-planar graphs is that planar graphs can be drawn on a 2-dimensional plane without any edge intersections, while non-planar graphs cannot. Non-planar graphs are more complex and can have edges that cross over each other when drawn on a plane.
Are there any limitations to planar graphs?
While planar graphs are useful in many applications, they do have some limitations. For example, they cannot accurately represent relationships that require crossing over edges, and they may not be able to represent certain types of networks with a large number of vertices and edges. In these cases, non-planar graphs may be a better representation.