Union and Intersection Operation On Graph

In graph theory, the data/objects belong to the same group but each piece of data differs from one other. In this article, we will see union and intersection operations on the graph. 

Union Operation  

Let G₁(V₁, E₁) and G₂(V₂, E₂) be two graphs as shown below in the diagram. The union of G₁ and G₂ is a graph G=G₁∪G₂, where vertex set V=V₁ ∪ V₂ and edge set E= E₁ ∪ E₂. 

For the above two graphs G₁ and G₂, we have vertices and edges as V₁= { A, B, C, D, E, F } and E₁= { 1,2,3,4,5,6,7 } and V₂= {B, C, D, E, G} and E₂= {3,6,8,9,10} respectively. 

So, in order to find the union of graphs G₁ and G₂, which can be denoted as G= G₁∪G₂. The vertex set of graph G will be V=V₁∪V₂= {A, B, C , D, E, F, G} and the edge set of graph G will be E=E₁∪E₂ ={1,2,3,4,5,6,7,8,9,10}. 

The resultant union graph G with all the vertices of set V and edges of set E will be as shown:    

Be careful to retain the original order of vertices and edges as in the original graph. 

Intersection Operation

Let G₁(V₁,E₁) and G₂(V₂,E₂) be two graphs. Then the intersection of G₁ and G₂ is a graph G=G₁∩G₂, whose vertex set V=V₁∩V₂  and edge set E=E₁∩E_2. 

For the above two graphs G1 and G2, we have vertices and edges as V1= { A, B, C, D, E } and E1= { 1,2,4,5,6,7 } and V2= {B, C, D, E, F} and E2= {3,5,7,8,9,10,11} respectively. 

So, in order to find the intersection of graphs G1 and G2, which can be denoted as G= G1∩G2. The vertex set of graph G will be V=V1∩V2= {B, C, D, E} and the edge set of graph G will be E=E1∩E2 ={3,5,7}. 

The resultant intersection graph G with all the vertices of set V and edges of set E will be as shown:

Be careful to retain the original order of vertices and edges as in the original graph in the resultant graph.