Operation's on Graph
Operations on Graphs
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Union of Graphs:
o
The union of two graphs, often denoted as G ∪
H, combines the vertex sets and edge sets of two graphs G and H to create a new
graph.
o
In the resulting graph, all vertices from both
original graphs are present, and the edges are the union of edges from both
graphs.
·
Intersection of Graphs:
o
The intersection of two graphs, often denoted as
G ∩ H, results in a new graph that contains only the common vertices and edges
shared by both graphs G and H.
·
Fusion of Graphs (Graph Fusion):
o
Vertex fusion, also known as vertex
identification or vertex contraction, is an operation in which two vertices,
let's call them V1 and V2, are replaced by a single new vertex, Vn.
o
As part of this operation, all edges incident to
either V1 or V2 are adjusted to connect to the new vertex Vn.
o
After the fusion, the resulting graph should
maintain the same connectivity properties and relationships as the original
graph, except that V1 and V2 are replaced by Vn.
·
Deletion of Graph Elements:
o
Deletion in graph theory can refer to the
removal of vertices and/or edges from a graph.
§
The specific operation depends on what you want
to delete.
o
Vertex deletion removes one or more vertices and
all edges incident to them.
o
Edge deletion removes specific edges while
leaving the vertices intact.
·
Ring Sum of Graphs (Ring Summation):
o
The "ring sum" of two graphs, often
denoted as G ⊕ H or G ⨁ H, is a graph operation that
combines two graphs by merging them along a common cycle (ring).
o
The result is obtained by adding edges between
corresponding vertices of the two graphs along their common cycle, creating a
new graph.