Planar graphs are a fundamental concept in graph theory, characterized by their ability to be drawn on a plane without edge crossings. This text delves into their distinctive properties, such as adhering to Euler's formula (v - e + f = 2) and having an edge limitation (3v - 6 edges). It also discusses techniques for recognizing planar graphs, the importance of Euler's formula, vertex colouring, and the Four Colour Theorem. Understanding these concepts is crucial for applications in network design, computational geometry, and algorithm development.
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Planar graphs can be represented on a plane without any edge intersections
Planar graphs are used in topology, network design, and computational geometry to understand the organization and interconnectivity of systems in two dimensions
A graph is considered planar if it can be embedded in a plane without edge crossings
Planar graphs divide the plane into distinct regions, including one unbounded exterior region and several bounded interior regions
Euler's formula states that for any connected planar graph with v vertices, e edges, and f faces, the relationship v - e + f = 2 must hold
Planar graphs are subject to an edge limitation, where a graph with v vertices can have at most 3v - 6 edges
To determine if a graph is planar, one may simplify the graph by eliminating loops and parallel edges
Kuratowski's Theorem states that a graph is non-planar if and only if it contains a subgraph that is homeomorphic to K5 or K3,3
If a graph does not contain subgraphs homeomorphic to K5 or K3,3, it may be redrawn to achieve a crossing-free representation
The planarity of a graph can impact the complexity and execution of algorithms in computational geometry and algorithm design
Planar graphs are instrumental in the development of network design and the study of polyhedra
Vertex colouring, where colours are assigned to vertices so that no two adjacent vertices are the same colour, has practical implications in areas such as scheduling and frequency allocation
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Planar graph representation
Planar graphs can be drawn on a plane without edge crossings, except at endpoints.
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Planar graph applications
Used in topology, network design, computational geometry for 2D system modeling.
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Planar graph embedding
Embedding is mapping a graph to a plane, edges intersect only at nodes.
