Simple Graphs: 99 Isomorphism 67 306 99 145 306 67 - - PowerPoint PPT Presentation

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Simple Graphs: 99 Isomorphism 67 306 99 145 306 67 - - PowerPoint PPT Presentation

The Graph Abstraction Mathematics for Computer Science MIT 6.042J/18.062J Same graph (different layouts) 122 257 122 145 257 Simple Graphs: 99 Isomorphism 67 306 99 145 306 67 isomorphism.1 isomorphism.2 Albert R Meyer April 1,


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Albert R Meyer April 1, 2013

isomorphism.1

Mathematics for Computer Science

MIT 6.042J/18.062J

Simple Graphs: Isomorphism

Albert R Meyer April 1, 2013

isomorphism.2

The Graph Abstraction

257 67 99 145 306 122 257 67 99 145 306 122

Same graph (different layouts)

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257 67 99 145 306 122 Albert Christos Jessica Sharat Sonya Grant

Same graph (different labels) The Graph Abstraction

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All that matters are the connections: graphs with the same connections are isomorphic

The Graph Abstraction

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Albert R Meyer April 1, 2013

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Isomorphism

two graphs are isomorphic when there is an edge-preserving matching

  • f their vertices.

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Are these isomorphic? Dog Pig Cat Cow Beef Tuna Corn Hay

f(Dog) = Beef f(Cat) = Tuna f(Cow) = Hay f(Pig) = Corn

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Edges preserved? Dog Pig Cat Cow Beef Tuna Corn Hay

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Dog Pig Cat Cow Beef Tuna Corn Hay Edges preserved? YES!

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Albert R Meyer April 1, 2013

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Dog Pig Cat Cow Beef Tuna Corn Hay Nonedges preserved?

YES!

isomorphic!

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Formal Def of Graph Isomorphism

G1 isomorphic to G2 means edge-preserving vertex matching:

∃ bijection f:V1 V2 with

u—v in E1 IFF f(u)—f(v) in E2

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degree 2 all degree 3 Nonisomorphism

Albert R Meyer April 1, 2013

Proving nonisomorphism

If some property preserved by isomorphism differs for two graphs, then they’re not isomorphic: # of nodes, # of edges, degree distributions, ….

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Finding an isomorphism?

many possible mappings: large search can use properties preserved by isomorphisms as a guide, for example:

  • a deg 4 vertex adjacent to a deg 3

can only match with

  • a deg 4 vertex also adjacent to a deg 3

but even so…

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Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013

Are these two graphs isomorphic?

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...nothing known is sure to be much faster than searching thru all bijections for an isomorphism

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6.042J / 18.062J Mathematics for Computer Science

Spring 2015 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.