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Chemistry
Political Science
Mathematics Traffic Control
PhysicsEconomics
IndustryBiology
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Australian National University
Social Sciences
WilliamtownBarnato
Cobar
Wollongong
Byrock
Nyngan
Walgett
Coonamble
Dubbo
Cowra
Sydney
Newcastle
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Armidale
GilgandraTamworth
and Generate Your Objects
Model Your Problems with Graphs
Engineering
Narjess Afzaly
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Modeling the problems in terms of graphs and producing therelevant graphs with computer in search for the best solution:
I Avoiding the real experimentsI Saving time, money and other resourcesI Applications in science and Industry
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The main challenge in graph generation is avoidingisomorphic copies.
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I A graph with 10 vertices can have more than two millionsisomorphic copies.
I Canonical Labeling: assigning a unique representativegraph to each isomorphic class of graphs.
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The Problem of Graph Generation
To make a complete list of non-isomorphic graphs in a givenclass.
Methods of generation differ in:I The Algorithm to generate each graph andI The method to avoid isomorphic copies.
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The Search Tree
A larger graph (child) is generated from a smaller graph(parent) by an operation (extension).
G5G6
G3
G1
G2
G6G5
G5
G13G12 G10 G10 G11 G10 G13 G12
G7 G8 G6G9
G2 G4
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dad
mom
baby
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Orderly Generation
I Only graphs in their canonical forms are accepted.
I The definitions of the extensions and the canonical formsmust be compatible.
G5G6
G3
G1
G2
G6G5
G5
G13G12 G10 G10 G11 G10 G13 G12
G7 G8 G6G9
G2 G4
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Generation by Canonical Construction Path
I Upper object: The graph to a childI Lower object: The graph to a parentI Reduction: the inverse of an extension
b
Extension
For each graph:I one specific lower object is defined as the base.I The winning lower objects are those in the same orbit as
the base.I A reduction is genuine if it reduces a winner.
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Generation by Canonical Construction Path
I Avoiding equivalent extensions for each graph.
I When a graph is generated, it is accepted only if it hasbeen generated through an extension whose inverseoperation is a genuine reduction.
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Comparing OG and GCCP
I In OG, graphs are accepted in their canonical form
I In GCCP graphs are accepted in a canonical way (on theCanonical Construction Path)
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The Software nauty
The software nauty is a set of procedures developed byMcKay that can calculate a canonically-labelled isomorphof the graph.
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Our Current Projects
I Generation of 4-regular Graphs,
I Generation of Principal Graph Pairs,
I Generation of Extremal Graphs Avoiding Cycles and
I Introducing a new canonical labeling that helps combiningOG and GCCP
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Your Projects
I Are you working on an interesting problem?
I Have you thought of modelling your problem with graphs?
I Can the current methods of generation help with yourprojects?
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Thank You!