Homomorphism

Two graphs G₁ and G₂ are said to be homomorphic, if each of these graphs can be obtained from the same graph 'G' by dividing some edges of G with more vertices. Take a look at the following example −

Divide the edge 'rs' into two edges by adding one vertex.

The graphs shown below are homomorphic to the first graph.

If G₁ is isomorphic to G₂, then G is homeomorphic to G₂ but the converse need not be true.

• Any graph with 4 or less vertices is planar.

• Any graph with 8 or less edges is planar.

• A complete graph K_n is planar if and only if n ≤ 4.

• The complete bipartite graph K_{m, n} is planar if and only if m ≤ 2 or n ≤ 2.

• A simple non-planar graph with minimum number of vertices is the complete graph K₅.

• The simple non-planar graph with minimum number of edges is K_{3, 3}.

Polyhedral graph

A simple connected planar graph is called a polyhedral graph if the degree of each vertex is ≥ 3, i.e., deg(V) ≥ 3 ∀ V ? G.

• 3|V| ≤ 2|E|
• 3|R| ≤ 2|E|