A graph is planar if it can be drawn on a plane without edge crossings. The complete graphs K5 and K3,3 are non-planar as they contain subgraphs that cannot be drawn without crossings. The Euler formula relates the number of vertices, edges, and faces in a planar graph as e - n + f = 2. Planarity can be tested using Kuratowski's theorem which states that a graph is planar unless it contains K5 or K3,3 as a subgraph.

4. ƒ( a ) = 1 ƒ( b ) = 6 ƒ( c ) = 8 ƒ( d ) = 3 ƒ( g ) = 5 ƒ( h ) = 2 ƒ( i ) = 4 ƒ( j ) = 7 An isomorphism between G and H Graph H Graph G

11. ƒ( a ) = 1 ƒ( b ) = 6 ƒ( c ) = 8 ƒ( d ) = 3 ƒ( g ) = 5 ƒ( h ) = 2 ƒ( i ) = 4 ƒ( j ) = 7 An isomorphism between G and H Graph H Graph G