Narjess Afzaly: Model Your Problem with Graphs and Generate your objects
- 1. Chemistry Political Science Mathematics Traffic Control Facebook Physics Economics Google Industry Biology C H C C C C O O H C H H H O H C Linguistics Australian National University Social Sciences WilliamtownBarnato Cobar Wollongong Byrock Nyngan Walgett Coonamble Dubbo Cowra Sydney Newcastle Bathurst Armidale Gilgandra Tamworth and Generate Your Objects Model Your Problems with Graphs Engineering Narjess Afzaly
- 2. Modeling the problems in terms of graphs and producing the relevant graphs with computer in search for the best solution: Avoiding the real experiments Saving time, money and other resources Applications in science and Industry
- 3. The main challenge in graph generation is avoiding isomorphic copies.
- 4. 4 21 3 4 1 2 3 A graph with 10 vertices can have more than two millions isomorphic copies. Canonical Labeling: assigning a unique representative graph to each isomorphic class of graphs.
- 5. The Problem of Graph Generation To make a complete list of non-isomorphic graphs in a given class. Methods of generation differ in: The Algorithm to generate each graph and The method to avoid isomorphic copies.
- 6. The Search Tree A larger graph (child) is generated from a smaller graph (parent) by an operation (extension). G5G6 G3 G1 G2 G6 G5 G5 G13G12 G10 G10 G11 G10 G13 G12 G7 G8 G6G9 G2 G4 + dad mom baby
- 7. Orderly Generation Only graphs in their canonical forms are accepted. The definitions of the extensions and the canonical forms must be compatible. G5G6 G3 G1 G2 G6 G5 G5 G13G12 G10 G10 G11 G10 G13 G12 G7 G8 G6G9 G2 G4
- 8. Generation by Canonical Construction Path Upper object: The graph to a child Lower object: The graph to a parent Reduction: the inverse of an extension b Extension For each graph: one specific lower object is defined as the base. The winning lower objects are those in the same orbit as the base. A reduction is genuine if it reduces a winner.
- 9. Generation by Canonical Construction Path Avoiding equivalent extensions for each graph. When a graph is generated, it is accepted only if it has been generated through an extension whose inverse operation is a genuine reduction.
- 10. Comparing OG and GCCP In OG, graphs are accepted in their canonical form In GCCP graphs are accepted in a canonical way (on the Canonical Construction Path)
- 11. The Software nauty The software nauty is a set of procedures developed by McKay that can calculate a canonically-labelled isomorph of the graph.
- 12. Our Current Projects Generation of 4-regular Graphs, Generation of Principal Graph Pairs, Generation of Extremal Graphs Avoiding Cycles and Introducing a new canonical labeling that helps combining OG and GCCP
- 13. Your Projects Are you working on an interesting problem? Have you thought of modelling your problem with graphs? Can the current methods of generation help with your projects?
- 14. Thank You!