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Chemistry
Political Science
Mathematics Traffic Control
PhysicsEconomics
IndustryBiology
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OH
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Linguistics
Australian National University
Social Sciences
WilliamtownBarnato
Cobar
Wollongong
Byrock
Nyngan
Walgett
Coonamble
Dubbo
Cowra
Sydney
Newcastle
Bathurst
Armidale
GilgandraTamworth
and Generate Your Objects
Model Your Problems with Graphs
Engineering
Narjess Afzaly
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
The main challenge in graph generation is avoidingisomorphic copies.
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21
3 4
1 2
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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.
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.
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
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
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.
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.
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)
The Software nauty
The software nauty is a set of procedures developed byMcKay that can calculate a canonically-labelled isomorphof the graph.
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
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?
Thank You!