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Two graphs are said to be isomorhic (to each other) if there is a 1-1 correspondence between their vertices and between their edges such that incidence relationship is preserved.
Note:Efficient isomorphism algorithms exist for certain type of graphs,such as Trees,Planar graphs,etc.
Necessary conditions:
1.The same number of vertices.
2.The same number of edges.
3.An equal number of vertices with a given degree.
Open Research Problem: to find a simple and efficient criterion for detection of isomorphism.
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
5.
Operations on Graphs G 1 =(V 1 , E 1 ) & G 2 =(V 2 , E 2 )
Union of two graphs
G=(V, E), with
V =V 1 V 2 , E =E 1 E 2
Ring Sum of two Graphs:G = G 1 + G 2
with V = V 1 V 2 , and
E =E 1 ∆ E 2
Intersection of two graphs, G=(V, E), with
V=V 1 V 2 & E =E 1 E 2
Deletion
Vertex deletion leads to deletion of all edges incident on it
Edge deletion does not imply deletion of end vertices.
A pair of vertices a , b in a graph are said to be fused if the two vertices are replaced by a single new vertex such that every edge that is incident on either a or b or on both is incident on the new vertex.
Note: The fusion of vertices does not alter the no of edges ,but it reduces the no of vertices by one.
2.Choose an edge e =vw of G such that w is not v and e has the smallest weight among the edges of G incident with v
3.If edges have been chosen involving end-points v,w,….t, choose an edge f = mk with m in S{v,w,….t} and k not in this setS such that f has the smallest weight among the edges of G with precisely one end in S.
4.Stop after n – 1 edges have been chosen. Otherwise repeat step3.
Two graphs are said to be homeomorphic if one graph can be obtained from the other by the creation of edges in series(ie by insertion of vertices of degree 2) or by the merger of edges in series.
Planarity test:
a necessary and sufficient condition for a graph to be planar is that G does not contain either of the Kuratowski’s graphs K 5 or K 3,3 or
any graph homeomorhic to either of them
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
12.
Operations on Graphs G 1 =(V 1 , E 1 ) & G 2 =(V 2 , E 2 )
Union of two graphs
G=(V, E), with
V =V 1 V 2 , E =E 1 E 2
Ring Sum of two Graphs:G = G 1 + G 2
with V = V 1 V 2 , and
E =E 1 ∆ E 2
Intersection of two graphs, G=(V, E), with
V=V 1 V 2 & E =E 1 E 2
Deletion
Vertex deletion leads to deletion of all edges incident on it
Edge deletion does not imply deletion of end vertices.
A pair of vertices a , b in a graph are said to be fused if the two vertices are replaced by a single new vertex such that every edge that is incident on either a or b or on both is incident on the new vertex.
Note: The fusion of vertices does not alter the no of edges ,but it reduces the no of vertices by one.
Graph G is planar if there exists some geometric representation of G which can be drawn on a plane with no cross-over of its edges.Otherwise, G is non-planar.
The complete graphs K 5 and K 3,3 are non-planar; K 3 is planar
Euler Formula for planar graph(n,e) is there are e – n + 2 regions
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