The document discusses graphs and their applications. It defines key graph terms like vertices, edges, directed/undirected graphs, paths, cycles, etc. It then describes algorithms for finding minimum spanning trees, Eulerian cycles, Hamiltonian paths, and approximations for the traveling salesman problem. Examples are provided to illustrate minimum spanning tree and Christofide's algorithms for TSP.

1. INTERESTING APPLICATIONS OF
GRAPHS
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2.  Graph.
 Vertex.
 Edge.
 Undirected Graph.
 Directed Graph.
 Path.
 Cycle.
 Eulerian Cycle and
 Hamiltonian Cycle.
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3.  A graph G consists of a finite set of ordered pairs, called edges
E, of certain entities called vertices V. Edges are also called as
arcs or links. Vertices are also called as nodes or points.
 G=(V,E)
 A graph is a set of vertices and edges. A vertex may represent
a state or a condition while the edge may represent a relation
between two vertices.
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4.  Directed Graph
 Directed Graphs or DIGRAPHS make reference to edges which are
directed (i.e.) edges which are ordered pairs of vertices.
 Undirected Graph
 A graph whose definition makes reference to unordered pairs of
vertices as edges is known as an undirected graph
 Path
 A simple path is a path in which all the vertices except possibly the
first and last vertices are distinct
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5.  Cycle
 A cycle is a simple path in which the first and last vertices are the same
. A cycle is also known as a circuit, elementary cycle, circular path or
polygon.
 Eulerian Graph
 A walk starting at any vertex going through each edge exactly once
and terminating at the start vertex is called an Eulerian walk or line.
 A Hamiltonian path in a graph is a path that visits each vertex
in the graph exactly once.
 A Hamiltonian cycle is a cycle that visits each vertex in the
graph exactly once and returns to the starting vertex.
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6.  Let G = (V, E) be an undirected connected graph. A subgraph
T = (V, E’) of G is a spanning tree of G iff T is a tree
 Given G = (V, E) to be a connected, weighted undirected graph
where each edge involves a cost, the extraction of a spanning
tree extends itself to the extraction of a minimum cost
spanning tree.
 A minimum cost spanning tree is a spanning tree which has a
minimum total cost.
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7. 2 24 3 2 3
4 4
1 1
23 9 9
6 18 6
6 6
5 4 5 4
16 11 11
8 5 8 5
7 7
10 14
7 21 8 7 8
G = (V, E) T = (V, F) w(T) = 50
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8.  Select an arbitrary node as the initial tree (T)
 Augment T in an iterative fashion by adding the outgoing edge
(u,v), (i.e., u  T and v  G-T ) with minimum cost (i.e.,
weight)
 The algorithm stops after |V | - 1 iterations
 Computational complexity = O (|V|2)
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9.  The algorithm then finds, at each stage, a new vertex to add
to the tree by choosing the edge (u, v) such that the cost of (u,
v) is the smallest among all edges where u is in the tree and v
is not.
 Algorithm would build the minimum spanning tree, starting
from v1. Initially, v1 is in the tree as a root with no edges.
 Each step adds one edge and one vertex to the tree.
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10. procedure PRIM(G)
E’ = ; /* Initialize E’ to a null set */
Select a minimum cost edge (u,v) from E;
V’ = {u}  Include u in V’ 
while V’ not equal to V do
Let (u, v) be the lowest cost edge such that u is in V’ and v is in V
– V’;
Add edge (u,v) to set E’;
Add v to set V’;
endwhile
end PRIM
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11. V2 3 V3
1
2
3 V6
V1 1
4
1 4
V4 V5
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12. V2
1
V1
V1
Algorithm starts
After the 1st
iteration
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13. V3
V2 3
V2 V3
3 1
3
1 1
3 V
V1
1
After the 2nd V5
iteration
After the 3rd
iteration
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14. V3
V2 3 V2 3 V3
1 1
2
1 1
V1 V1
1 V6
1
V4 V5 V4 V5
After the 4th After the 5th
iteration iteration
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15.  The Travelling Salesman Problem (TSP) is an NP-hard problem
studied in OR(Operational Research) and theoretical
computer science.
 You are given a set of n cities
 You are given the distances between the cities.
 You start and terminate your tour at your home city.
 You must visit each other city exactly once.
 Your mission is to determine the shortest tour.
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16.  The TSP is easy to state, takes no math background to
understand, and no great talent to find feasible solutions. It’s
fun, and invites recreational problem solvers.
 It inturn has many practical applications.
 Many hard problems, such as job shop scheduling, can be
converted algebraically to and from the TSP.
 A good TSP algorithm will be good for other hard problems.
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17.  Nearest Neighbour Algorithm.
 Lower Bound Algorithm.
 Tour Improvement Algorithm.
 Christofide’s Algorithm.
Etc.,
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18.  It works only on undirected graphs.
 Step 1, minimum spanning tree: we construct a minimum cost
spanning tree T for graph G
 Step 2, Creating a Cycle:Now that we have our MST M , we
can create a cycle W from it. In order to do this, we walk
along the nodes in a depth first search, revisiting vertices as
ascend. Graphically, this means outlining the tree.
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19.  Step 3, Removing Redundant Visits: In order to create a
plausible solution for the TSP, we must visit vertices exactly
once.
Since we used an MST, we know that each vertex is visited at
least once, so we need only remove duplicates in such a way
that does not increase the weight.
 Algorithm complexity is O(n3).
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20. Example: G(V,E):
6 Belmont
Arlington
7 4
3
8
Fantsila 4
Cambridge
5
8
Everett 7
6
Denmolt
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21. Minimum Hamiltonian Cycle
6 B
A
7 4
3
8
F 4
C
5
8
E 7
6
Cost = 34
D
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22. Step 1:
6 B
A
4
3
F
C
5
E
6
D
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23. Step 2:
6 B
A
4
3
F
C
5
E
6
Cycle: ABCBFEDEFBA Cost : 48
D
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24. Step 3:
6 B
A
4
3
F
C
5
E
6
Shortcut: FBA FA Saving : 2
D
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25. Step 3:
6 B
A
4
3
8
F
C
5
E
6
Shortcut: EFA EA Saving : 4
D
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26. Step 3:
6 B
A
4
3
8
F
C
8
E
6
Shortcut: FED FD Saving : 3
D
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27. Step 3:
6 B
A
4
8
4
F
C
8
E
6
Shortcut: CBF CF Saving : 3
D
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28. End:
6 B
A
4
8
4
F
C
8
E
6
Cycle : ABCFDEA
Cost : 36 D
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29. Knight’s Path:
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30.  Applying graph theory to a system means using a graph-
theoretic representation
 Representing a problem as a graph can provide a different
point of view.
 Representing a problem as a graph can make a problem much
simpler.
 More accurately, it can provide the appropriate tools for solving the
problem.
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