Graphs and Isomorphisms

Transcription

1 Graphs and Isomorphisms Discrete Structures (CS 173) Backyards of Old Houses in Antwerp in the Snow Van Gogh Madhusudan Parthasarathy, University of Illinois

2 Proof techniques: Direct Contrapositive Disproving Induction Contradiction Pigeon-hole principle 2-way bounding Discrete structures: Numbers Sets, functions, rels Graphs Trees 2

3 Fastest path from Chicago to Bloomington?

4 Fastest path from Chicago to Bloomington?

5 Fastest path from Chicago to Bloomington? start end 4

6 Fastest path from Chicago to Bloomington? start C end B 3

7 Other applications of graphs Modeling the flow of a network Traffic, water in pipes, bandwidth in computer networks, etc. 7

8 Basics of graphs Graph = (V, E) Terminology: vertex/node, edge, neighbor/adjacent, directed vs. undirected, simple graph, degree of a node overhead 8

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10 Degrees and handshaking theorem Loops count twice overhead 10

11 Types of graphs K n : complete graph with n nodes How many edges does each type have? overhead 11

12 Types of graphs C n : cycle graph with n nodes W n : wheel graph with n + 1 nodes How many edges does each type have? overhead 12

13 Subgraphs 13

14 Isomorphism An isomorphism from G 1 = (V 1, E 1 ) to G 2 = (V 2, E 2 ) is a bijection f: V 1 V 2 such that any pair of nodes a and b are joined by an edge iff f(a) and f b are joined by an edge Two graphs are isomorphic if there is an isomorphism between them overhead 14

15 Isomorphism examples An isomorphism from G 1 = (V 1, E 1 ) to G 2 = (V 2, E 2 ) is a bijection f: V 1 V 2 such that any pair of nodes a and b are joined by an edge iff f(a) and f b are joined by an edge overhead 15

16 Isomorphism examples An isomorphism from G 1 = (V 1, E 1 ) to G 2 = (V 2, E 2 ) is a bijection f: V 1 V 2 such that any pair of nodes a and b are joined by an edge iff f(a) and f b are joined by an edge overhead 16

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18 Necessary but not sufficient requirements for two graphs to be isomorphic overhead 18

19 Requirements for two graphs to be isomorphic Same number of nodes and edges Same number of nodes of degree k Every subgraph in one must have a ``matching subgraph (isomorphic subgraph) in the other 19

20 Isomorphism is an equivalence relation: reflexive, symmetric, and transitive 20

21 Automorphism: an isomorphism from a graph to itself Automorphisms identify symmetries in the graph How many different automorphisms? C 4 overhead 21

22 Bridges of Konigsberg Possible to cross all bridges exactly one and end up where you started? First theorem of graph theory! Image source: 22

23 Terminology of walks walk: sequence of connected nodes/edges closed walk: start and end point are the same path: walk with no node used more than once c f a b d e overhead 23

24 Terminology of walks cycle: closed walk with 3+ nodes, no nodes except the first/last used more than once acyclic: graph without cycles Euler circuit: closed walk that uses each edge exactly once b c f a d overhead e 24

25 Bridges of Konigsberg Possible to cross all bridges exactly one and end up where you started? overhead 25

26 Connected graphs 26

27 More connectivity terminology The distance of two nodes is the minimum number of edges that forms a walk from one node to the other. The diameter of a graph is the maximum distance between any two nodes in a graph. 27

28 More connectivity terminology A graph is connected if there is a walk between any two nodes. A connected component is a subset of nodes that are connected to each other but not connected to any other nodes. overhead 28

29 Connected components In fact, define u R v iff there is a walk from u to v. R is reflexive, symmetric, and transitive. an equivalence relation. The equivalence classes of R are. 29

30 Bipartite graphs A graph G=(V,E) is bipartite if we can partition the set of vertices into two (disjoint) sets V1 and V2 such that all edges are between a vertex in V1 and a vertex in V2 (i.e., no edge should be between two vertices of V1, and no edge should be between two vertices of V2). 30

31 k-colorability A graph is k-colorable if you can assign a color from {1, k} to every vertex such that every pair of adjacent vertices get different colors. 31

32 2-colorability and bipartite A graph is 2-colorable iff it is bipartite. 32

33 Colorability Every graph with maximum degree k is (k+1)- colorable. 33

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