Introduction to Graph Theory
Sonia Toubaline
Contact: [email protected]
Humanitrian Logistic pre-course
O BJECTIVES
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Graph modelling
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Basic notions and definitions in graph theory
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Examples
D ECISION M AKING
Definition
Scientific approach aiming to help any organisation in the elaboration of
its decisions
Objective
Model certain types of problems using mathematical tools and solve
them with known and automated solution methods
D ECISION M AKING P ROCESS
⇒ Graph modelling
G RAPH MODELLING
Applications
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Communication network : internet, mobile phone, . . .
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Transportation network: road, underground, plane, . . .
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Critical infrastructures: power plants, . . .
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Electricity network
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Water distribution network
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Facility location: warehouse location, . . .
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Security
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Humanitarian logistic: transshipment, scheduling, . . .
Interest: A simple representation of the structure of diverse situations
or systems
C OMMUNICATION NETWORKS
Problem: route all the messages, task assignment and process
C OMMUNICATION NETWORKS
⇒
Graph modelling
Entities: computers, cables, switches,. . .
Node: computers, switches ⇒ N set of all the nodes
Edge: cables
2 extremity nodes and connection between them ⇒ E set of all the edges
⇒ non oriented graph G = (N, E)
T RANSPORT NETWORKS
Problem: commodity delivery, task scheduling, shortest path, minimise
the cost, minimise the ending time of the delivery, . . .
T RANSPORT NETWORKS
⇒
Graph modelling
Entities: trucks, houses, factories, depot centres, warehouses, roads, . . .
Node: different locations ⇒ N set of all the nodes
Arc: roads
- 2 extremity nodes and connection between them
- Initial extremity node, terminal extremity node
⇒ A set of all the arcs
⇒ an oriented graph G = (N, A)
PARISIAN U NDERGROUND
W EIGHTED GRAPHS
⇒
A graph G = (N, E)
Weights associated to nodes & edges: cost of communication, throughput,
capacity cables,. . .
W EIGHTED GRAPHS
⇒
A graph G = (N, A)
Weights associated to nodes or arcs: distance, travel time, quantity of fuel
used, amount of loads, . . .
A IRLINE FLIGHT NETWORK
Problem: flight scheduling, ticket prices, planes assignment . . .
⇒
Oriented graph G = (N, A)
Nodes: countries, cities
Arcs: links/ flights/ connections
Weights associated to arcs (e.g. distance, time travel, quantity of fuel used, . . .)
& nodes (e.g.passenger number, plane capacity, amount of commodities,. . .)
E LECTRICITY NETWORK - S MART G RIDS
Problem: real time observability
⇒
Oriented graph G = (N, A)
Nodes: plants, stations, houses
Arcs: links/ cables/ connections
Weights associated to arcs (e.g.cable lengths, cable characteristics . . .) & nodes
(e.g. voltages, angles,. . .)
WATER DISTRIBUTION NETWORK
Problem: sensor placement for contamination detection
⇒
Oriented graph G = (N, A)
Nodes: Houses, companies, tanks, sources, . . .
Arcs: pipelines
Weights associated to arcs (e.g. pipeline bounds/capacities, pipelines length
. . .) & nodes (e.g. demand, production/storage,. . .)
B ASIC NOTIONS AND DEFINITIONS -S UMMARY
Weighted (non) oriented graph G = (N, A) (G = (N, E))
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N : set of nodes
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E: set of edges ; A est of arcs
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Edges/arcs: characterises the existence of a possibility
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Edge: 2 adjacent nodes. These nodes are incident to that edge.
(Extremity nodes are adjacent and incident to the edge)
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Arc: initial and terminal extremity nodes
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Weight: values associated to edges, arcs and/or nodes.
A DJACENCY
Neighbours of a node: set of all its adjacent nodes
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A graph can be defined by set of nodes and set of neighbours.
PATH AND CYCLE
Path : succession of adjacent arcs (edges)
Cycle: a path whose initial and terminal extremity are the same.