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Graph Theory: Lecture No. 1
Graph Theory: Lecture No. 1
L. Sunil Chandran
Computer Science and Automation,Indian Institute of Science, BangaloreEmail: [email protected]
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Graph Theory: Lecture No. 1
References
1 Reinhard Diestel : Graph Theory (Springer)
2 Douglas B. West: Introduction to Graph Theory(Prentice-Hall India)
3 A. Bondy and U.S.R. Murty: Graph Theory (Springer)
4 B. Bollabas : Modern Graph Theory (Springer)
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Graph Theory: Lecture No. 1
What is a Graph ?
It is a triple consisting of a vertex set V (G ), an edge set E (G ) anda relation that associates with each edge two vertices (notnecessarily distinct) called its end points.
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Graph Theory: Lecture No. 1
A loop
Multiple edge.
Simple Graph
Finite Graph
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Graph Theory: Lecture No. 1
Some simple graphs
Complete Graph.
Cycle
Path
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Graph Theory: Lecture No. 1
Subgraph and Induced Subgraph
H is a subgraph of G : Then V (H) ⊆ V (G ), E (H) ⊆ E (G ).The assignment of end points to edges in H is the same asthat in G .
H is an induced subgraph of G on S , where S ⊆ V (G ): ThenV (H) = S and E (H) is the set of edges of G such that boththe end points belong to S .
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Graph Theory: Lecture No. 1
A graph G is connected if each pair of vertices belongs to a path.
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Graph Theory: Lecture No. 1
Isomorphism
An Isomorphism from a simple graph G to a simple graph H is abijection f : V (G ) → V (H) such that (u, v) ∈ E (G ) if and only if(f (u), f (v)) ∈ E (H)
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Graph Theory: Lecture No. 1
Forest and Tree
A graph without any cycle is acyclic.
A forest is an acyclic graph
A tree is a connected acyclic graph.
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Graph Theory: Lecture No. 1
Bipartite Graph
A graph G is bipartite if V (G ) is the union of two disjoint(possibly empty) independent sets called partite sets of G .(A subset S ⊆ V (G ) is an independent set if the induced subgraphon S contains no edges. )
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Graph Theory: Lecture No. 1
A tree is a bipartite graph.
Can we say that the complete graph Kn is bipartite ?
The complete bipartite graph.
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Graph Theory: Lecture No. 1
Vertex Cover
A set S ⊆ V (G ) is a vertex cover of G (of the edges of G ) if everyedge of G is incident with a vertex in S .A vertex cover of the minimum cardinality is called a minimumvertex cover. We will denote this set by MVC(G).
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Graph Theory: Lecture No. 1
What is the cardinality of MVC in the complete graph Kn ?
What about the complete bipartite graph Km,n ?
The cycle Cn, when n is even and odd ?
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Graph Theory: Lecture No. 1
The cardinality of a biggest independent set in G is called theindependence number (or stability number) of G and is denoted byα(G ).
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Graph Theory: Lecture No. 1
Is there any relation between |MVC (G )| and α(G ) ?
If we remove a VC from G , the rest is an independent set.
So, if we remove MVC from G , the rest, i.e. V −MVC is anindependent set.
So, α(G ) ≥ n − |MVC (G )|. Thus |MVC (G )| ≥ n − α(G ).
Similiarly if we remove any independent set from G , the restis VC, and so |MVC | ≤ n − α(G ).
Thus we get |MVC | = n − α(G ).
If we denote —MVC(G)— by β(G ), then we haveβ(G ) + α(G ) = n.
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Graph Theory: Lecture No. 1
Matching
A set M of independent edges in a graph is called a matching.
M is a matching of U ⊆ V (G ), if every vertex in U is incidentwith an edge in M.
Then a vertex in U is a matched vertex. The vertices whichare not incident with any edge of M is unmatched.
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Graph Theory: Lecture No. 1
A matching M is a perfect matching of G , if every vertex in G ismatched by M.If G has n vertices, what is the cardinality of a perfect matching Mof G ?
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Graph Theory: Lecture No. 1
The cardinality of the biggest matching in G can be denoted byα′(G ).
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Graph Theory: Lecture No. 1
What is the value of α′(G ) for:
Cycle Cn
Path Pn
Complete Graph Kn
Complete Bipartite graph Km,n