What are Planar Graphs? | Graph Theory

What are planar graphs? How can we draw them in the plane? In today's graph theory lesson we'll be defining planar graphs, plane graphs, regions of plane graphs, boundaries of regions of plane graphs, and introducing Euler's formula for connected plane graphs.

A planar graph is a graph that can be drawn in the plane with no edge crossings. A plane graph is a planar graph that has been drawn in the plane with no edge crossings. Thus, a graph being planar is not dependent on how it is drawn, but only on how it CAN be drawn, whereas a graph being a plane graph is dependent on how it is drawn. The complete graph K4 can be drawn as a plane graph and is thus planar, but can also be drawn not as a plane graph.

A nonplanar graph is a graph that cannot be drawn in the plane without edge crossings. That is, a nonplanar graph will always have edge crossings if drawn in the plane. An example of a nonplanar graph is K3,3 the complete bipartite graph with two partite sets of cardinality 3.

A plane graph divides the plane into different regions. The subgraph, comprised of vertices and edges that are incident with a region, make up the boundary of the region. The boundary of a region is also sometimes defined (roughly) as the closed walk that encloses a region.

In a connected plane graph, the number of vertices (n), minus the number of edges (m), plus the number of regions (r), that is: n-m+r, is equal to 2. This is Euler's formula for connected plane graphs.

What are isomorphic graphs?    • What are Isomorphic Graphs? | Graph I...  
What are complete bipartite graphs?    • What are Complete Bipartite Graphs? |...  

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