Graph (1A) Young Won Lim 4/19/18

Transcription

1 Graph (1A)

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using LibreOffice and Octave.

3 Some class of graphs (1) Complete graph A complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges. Connected graph In an undirected graph, an unordered pair of vertices {x, y} is called connected if a path leads from x to y. Otherwise, the unordered pair is called disconnected. Bipartite graph A bipartite graph is a graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2. Graph (5A) 3

4 Complete Graphs Graph (5A) 4

5 Connected Graphs This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed. With vertex 0 this graph is disconnected, the rest of the graph is connected. Graph (5A) 5

6 Bipartite Graphs Example of a bipartite graph without cycles A complete bipartite graph with m = 5 and n = 3 A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. Graph (5A) 6

7 Complete Graphs Graph (5A) 7

8 Complete Bipartite Graphs Graph (5A) 8

9 Star Graphs Graph (5A) 9

10 Wheel Graphs Graph (5A) 10

11 Some class of graphs (2) Planar graph A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect. Cycle graph A cycle graph or circular graph of order n 3 is a graph in which the vertices can be listed in an order v1, v2,, vn such that the edges are the {vi, vi+1} where i = 1, 2,, n 1, plus the edge {vn, v1}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph. Tree A tree is a connected graph with no cycles. Graph (5A) 11

12 Planar Graphs A planar graph and its dual Graph (5A) 12

13 Cycle Graphs Graph (5A) 13

14 Tree Graphs Graph (5A) 14

15 Hypercube A hypercube can be defined by increasing the numbers of dimensions of a shape: 0 A point is a hypercube of dimension zero. 1 If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one. 2 If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square. 3 If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube. 4 If one moves the cube one unit length into the fourth dimension, it generates a 4- dimensional unit hypercube (a unit tesseract). Tesseract Graph (5A) 15

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17 Gray Code Tesseract Graph (5A) 16

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19 Adjacency Lists Graph (5A) 17

20 Incidence Matrix Graph (5A) 18

21 Adjacency Matrix Graph (5A) 19

22 Hamiltonian Path Graph (5A) 20

23 Minimum Spanning Tree Graph (5A) 21

24 Seven Bridges of Königsberg The problem was to devise a walk through the city that would cross each of those bridges once and only once. Graph (5A) 22

25 Shortest path problem Graph (5A) 23

26 Traveling salesman problem Graph (5A) 24

27 Euler Cycle Any connected graph with even degree vertices An Euler cycle A proof by induction on the number of edges in G A connected graph G with even degree vertices only and k edges (k < n) An Euler cycle Assume this is true A connected graph G with even degree vertices only and n edges An Euler cycle Then this holds true Johnsonbough, Discrete Mathematics Graph (5A) 25

28 Euler Cycle Any connected graph with even degree vertices which has n edge An Euler cycle Any connected graph with even degree vertices which has n-1 edge Any connected graph with even degree vertices which has n-2 edge An Euler cycle An Euler cycle Any connected graph with even degree vertices which has 2 edge Any connected graph with even degree vertices which has 1 edge An Euler cycle An Euler cycle Johnsonbough, Discrete Mathematics Graph (5A) 26

29 Euler Cycle Base Cases n = 0 edge n = 1 edge all even degree vertices n = 2 edge an Euler cycle Johnsonbough, Discrete Mathematics Graph (5A) 27

30 Euler Cycle decrease the number of edges by one A connected graph G with even degree vertices only and n edges (k < n) e 1 e 2 v 1 v 2 v 3 all even degree vertices P: a path from v to v 1 A connected graph G' with even degree vertices only and n-1 edges (k < n) P': a portion of the path P that are in G' e v 1 v 2 v 3 all even degree vertices Johnsonbough, Discrete Mathematics Graph (5A) 28

31 Euler Cycle a path from v to v 1 Case 1: P ends at v 1 Case 2: P ends at v 2 Case 3: P ends at v 3 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P P P v v v v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P ' P ' P ' v v v Johnsonbough, Discrete Mathematics Graph (5A) 29

32 Euler Cycle F B H E D C I J G ABCDEFGHIJK A K en.wikipedia.org Graph (5A) 30

33 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components Graph (5A) 31

34 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component Graph (5A) 32

35 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components Graph (5A) 33

36 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component Graph (5A) 34

37 References [1] [2]