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, 2013 Albert R Meyer April 1, 2013 The Graph Abstraction The Graph Abstraction Same graph (different labels) All that matters are the connections: 122 Grant 145 Sharat Albert 257 graphs with the 67 same connections Christos 306 Sonya are isomorphic 99 Jessica isomorphism.3 isomorphism.4 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 1

Isomorphism Are these isomorphic? two graphs are isomorphic Hay when there is an Dog Pig edge-preserving Corn matching Cow Cat Beef Tuna of their vertices. f(Dog) = Beef f(Cow) = Hay f(Cat) = Tuna f(Pig) = Corn isomorphism.5 isomorphism.6 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 Edges preserved? Edges preserved? YES! Hay Hay Dog Dog Pig Pig Corn Corn Cow Cow Cat Cat Beef Beef Tuna Tuna isomorphism.7 isomorphism.8 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 2

YES! Nonedges preserved? Formal Def of Graph Isomorphism G 1 isomorphic to G 2 means Hay edge-preserving vertex matching : Dog Pig ∃ bijection f:V 1 � V 2 with Corn u—v in E 1 IFF f(u)—f(v) in E 2 Cow Cat Beef Tuna isomorphic! isomorphism.9 isomorphism.10 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 Nonisomorphism Proving nonisomorphism If some property preserved by isomorphism differs for two graphs, then they’re not isomorphic: # of nodes, # of edges, degree distributions, … . degree 2 all degree 3 isomorphism.11 isomorphism.12 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 3

Are these two graphs isomorphic? 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 ...nothing known is sure to be • a deg 4 vertex also adjacent to a deg 3 much faster than searching thru but even so… all bijections for an isomorphism isomorphism.13 isomorphism.14 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 4

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