Chemistry

Political Science

Mathematics Traffic Control

Facebook

PhysicsEconomics

Google

IndustryBiology

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Linguistics

Australian National University

Social Sciences

WilliamtownBarnato

Cobar

Wollongong

Byrock

Nyngan

Walgett

Coonamble

Dubbo

Cowra

Sydney

Newcastle

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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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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).

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G2 G4

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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.

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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

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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!