are a fascinating subset of graphs that can be drawn without edge crossings. They have unique properties, like and maximum edge limits, that set them apart from other graph types.
Understanding planar graphs is crucial for real-world applications. From circuit design to map coloring, these concepts help solve complex problems. provides a powerful tool for determining graph planarity.
Planar Graphs and Their Properties
Properties of planar graphs
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Planar graphs drawn on plane without edge crossings edges intersect only at vertices
Every planar graph has subgraphs also planar
in planar graph with n vertices: 3n−6 (for n≥3)
Every planar graph 4-colorable ()
Dual graphs constructed by placing vertex in each face of original planar graph edges correspond to adjacent
Faces in planar graphs
Faces bounded by edges in planar embedding include outer (infinite) face
Each face bounded by cycle of edges number of surrounding edges
Connected planar graph faces may have faces with holes
Euler's Formula and Graph Planarity
Applications of Euler's formula
Euler's formula for planar graphs: [V](https://www.fiveableKeyTerm:v)−[E](https://www.fiveableKeyTerm:e)+[F](https://www.fiveableKeyTerm:f)=2 (V vertices, E edges, F faces)
Determine number of faces in planar graph
Calculate maximum edges in planar graph
Prove of certain graphs (K5 and K3,3)
Variations for graphs on other surfaces (torus) and multiple connected components
Kuratowski's theorem for planarity
Graph planar if and only if no subgraph homeomorphic to K5 or K3,3
graphs obtained by inserting or removing degree 2 vertices
Identify K5 or K3,3 subdivisions in graph to prove non-planarity
equivalent formulation using graph minors instead of homeomorphism