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# Isograph

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• 1. Distance and Centres in a Tree
• Distance between vertices is the shortest path between them.
• Eccentricity of a vertex is its distance from the vertex farthest to it in the graph
• A centre of a graph is a vertex with minimum eccentricity in the graph.
• The distance between vertices of a connected graph is a metric
• Every tree has either one or two centres and, in the latter case, they are adjacent.
• The eccentricity of a centre in a tree is called the radius of the tree.
• The diameter of a tree is the length of the longest path
• 2. Property of Trees
• 1.Any pair of vertices in a tree has exactly one path between them.
• 2.Conversely if G is connected and there is exactly one path between every pair of vertices, then G is a tree.
• 3.Suppose a tree has n i vertices of degree I for i = 1,2,..k, then
• n 1 = 2 +n 3 +2n 4 +…+(k – 2) n k
• 4.Every tree has either one or two centres.
• 3. Iso-morphism
• 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.
• 6. Operations on Graphs Fusion
• 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.
• 7. Prim’s Algorithm
• 1.Choose any vertex v of G
• 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.
• 8. Dual Graph
• Given graph G, construct its dual G * as follows:
• i)inside each face of G, choose a point v *
• ii)corresponding to each edge e of G we draw a line e * that crosses e only and joins the vertices v * in the faces adjoining e.
• Note: G * is the dual of G containing vertices v * and edges e *
• If G is connected with n vertices,e edges,f regions, then G * has
• f vertices, e edges, and n regions.
• . Neither K 5 nor K 3,3 has a dual
• 9. Detection of Planarity
• Elementary Reduction
• Two edges are in series if they have exactly one vertex in common and if this vertex is of degree 2.
• Step1: Consider a component of the Graph
• Step2:Remove self-loops
• Step3:Remove all but one edge between every pair of vertices.
• Step4:Eliminate a vertex of degree 2 by merging two edges in seres
• Apply step 3 & 4 repeatedly for a non-separable graph, reducing G to H, where
• i) H is a single edge, or
• ii) a complete graph of 4 vertices,K 4 or
• iii)a non-separable,simple graph with n>=5 & e>= 7
• 10. Kuratowski’s Theorems
• Homeomorphic Graphs :
• 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.
• 13. Operations on Graphs Fusion
• 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.
• 14.
• 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
• For connected planar graph e<= 3n -6 .
• & 2e >= 3f
Planar Graph