It contains almost every basic things necessary for understanding Network and Tree. It has a clear contents on entire Graph theory.

1. NETWORK ANDTREE
IN GRAPHTHEORY
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Presented by: The Camouflage
BSc.CSIT 2nd Semester
Gagan Puri
Rabin BK
Bikram Bhurtel
Prism Karki

2. Network flow
Spanning Tree
Tree traversal
References
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3. Also known as a transportation network
It is a directed graph where each edge has a capacity and each
edge receives a flow
Often in operations research, a directed graph is called a
network, the vertices are called nodes and the edges are called
arcs
Network Flow
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4. A network is a graph G = (V, E), where V are a set of
vertices and E are a set of edges
a non-negative function c: V × V → ℝ∞ is called the
capacity function
If two nodes in G are distinguished, a source s and a sink t,
then (G, c, s, t) is called a flow network
Definition
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5. Source : no incoming edges
Sink : no outgoing edges
Slack : ( capacity – flow )
Augmented path : a path (u1, u2, ..., uk), where u1 = s, uk
= t, and cf (ui, ui + 1) > 0
Saturated : if flow = capacity
Unsaturated : if flow < capacity
Basic terms
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6. Example: To find a maximum flow :
A network is at maximum flow if and only if there
is no augmenting path in the residual network Gf.
s → v → t
s → v → u → t
Fig: optimal solution 6

7. Used to model traffic in road system, circulation with diamonds, fluids in
pipes and anything similar in which sth travels through a network of nodes
Electrical distribution systems
Ecosystem network analysis
Maximum flow
Multi – commodity flow problem
Minimum cost flow problem
Circulation problem
Network with gains / generalised network
Source localization problem
Applications
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8. For a simple graph G, Spanning tree is a subgraph of G.
It contains every vertex of G.
All the vertices are connected
The connected edges must not form a cycle
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Spanning Tree
Simple graph Spanning tree
Not a tree since there
are small circuits and
closed cycles
It is a tree because
there is no closed cycle
and there is a single
path to every edge.

9. It is also a spanning tree
It has the smallest possible sum of weights of its edges
Algorithms used for constructing the minimum spanning tree
 Prim's Algorithm
 Kruskal's Algorithm
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Minimum spanning tree

10. Step 1: Select the vertex at random
Step 2: Find all the edges that connect the tree to
new vertices, find the minimum and add it
to the tree
Step 3: Keep repeating step 2 until we get a
minimum spanning tree
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Prim's AlgorithmPrim's Algorithm

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12. Step 1:Sort all the edges from low weight to high
Step 2: Take the edge with the lowest weight and
add it to the spanning tree. If adding the
edge created a cycle, then reject this edge
Step 3: Keep adding edges until all
vertices are not included
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Kruskal's Algorithm

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21. References
Textbook reference:
• Discrete Math and its application,Kenneth H. Rosen
Web Reference:
 http://scanftree.com/Data_Structure/prim%27s-algorithm
 https://www.cse.ust.hk/~dekai/271/notes/L07/L07.pdf
 http://www.tutorvista.com/content/math/prims-algorithm/
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22. Queries
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