Post on 20-Jun-2020
Young Won Lim4/19/18
Graph (1A)
Young Won Lim4/19/18
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Graph (5A) 3 Young Won Lim4/19/18
Some class of graphs (1)
https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)
Complete graphA complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.
Connected graphIn 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 graphA 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) 4 Young Won Lim4/19/18
Complete Graphs
https://en.wikipedia.org/wiki/Complete_graph
Graph (5A) 5 Young Won Lim4/19/18
Connected Graphs
https://en.wikipedia.org/wiki/Connectivity_(graph_theory)
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 isconnected.
Graph (5A) 6 Young Won Lim4/19/18
Bipartite Graphs
https://en.wikipedia.org/wiki/Bipartite_graph
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) 7 Young Won Lim4/19/18
Complete Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph (5A) 8 Young Won Lim4/19/18
Complete Bipartite Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph (5A) 9 Young Won Lim4/19/18
Star Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph (5A) 10 Young Won Lim4/19/18
Wheel Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph (5A) 11 Young Won Lim4/19/18
Some class of graphs (2)
Planar graphA planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.
Cycle graphA 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.
TreeA tree is a connected graph with no cycles.
https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)
Graph (5A) 12 Young Won Lim4/19/18
Planar Graphs
https://en.wikipedia.org/wiki/Planar_graph
A planar graph and its dual
Graph (5A) 13 Young Won Lim4/19/18
Cycle Graphs
https://en.wikipedia.org/wiki/Cycle_graphhttps://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph (5A) 14 Young Won Lim4/19/18
Tree Graphs
https://en.wikipedia.org/wiki/Cycle_graph
Graph (5A) 15 Young Won Lim4/19/18
Hypercube
https://en.wikipedia.org/wiki/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) 16 Young Won Lim4/19/18
Gray Code
https://en.wikipedia.org/wiki/Gray_code
Tesseract
Graph (5A) 17 Young Won Lim4/19/18
Adjacency Lists
https://en.wikipedia.org/wiki/Adjacency_list
Graph (5A) 18 Young Won Lim4/19/18
Incidence Matrix
https://en.wikipedia.org/wiki/Incidence_matrix
Graph (5A) 19 Young Won Lim4/19/18
Adjacency Matrix
https://en.wikipedia.org/wiki/Adjacency_matrix
Graph (5A) 20 Young Won Lim4/19/18
Hamiltonian Path
https://en.wikipedia.org/wiki/Path_(graph_theory)
Graph (5A) 21 Young Won Lim4/19/18
Minimum Spanning Tree
https://en.wikipedia.org/wiki/Minimum_spanning_tree
Graph (5A) 22 Young Won Lim4/19/18
Seven Bridges of Königsberg
https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
The problem was to devise a walk through the city that would cross each of those bridges once and only once.
Graph (5A) 23 Young Won Lim4/19/18
Shortest path problem
https://en.wikipedia.org/wiki/Shortest_path_problem
Graph (5A) 24 Young Won Lim4/19/18
Traveling salesman problem
https://en.wikipedia.org/wiki/Travelling_salesman_problem
Graph (5A) 25 Young Won Lim4/19/18
Euler Cycle
Johnsonbough, Discrete Mathematics
Any connected graph with even degree vertices
An Euler cycle
A proof by induction on the number of edges in G
A connected graph Gwith even degree vertices only and k edges (k < n)
A connected graph Gwith even degree vertices onlyand n edges
Assume this is trueAn Euler cycle
An Euler cycleThen this holds true
Graph (5A) 26 Young Won Lim4/19/18
Euler Cycle
Johnsonbough, Discrete Mathematics
Any connected graph with even degree verticeswhich has n edge
Any connected graph with even degree vertices which has n-1 edge
Any connected graph with even degree vertices which has n-2 edge
Any connected graph with even degree vertices which has 2 edge
Any connected graph with even degree verticeswhich has 1 edge
An Euler cycle
An Euler cycle
An Euler cycle
An Euler cycle
An Euler cycle
Graph (5A) 27 Young Won Lim4/19/18
Euler Cycle – Base Cases
Johnsonbough, Discrete Mathematics
all even degree vertices
n = 0 edge
n = 1 edge
n = 2 edge an Euler cycle
Graph (5A) 28 Young Won Lim4/19/18
Euler Cycle – decrease the number of edges by one
Johnsonbough, Discrete Mathematics
A connected graph Gwith even degree vertices only and n edges (k < n)
A connected graph G'with even degree vertices only and n-1 edges (k < n)
v1
v2
v3
v1
v2
v3
all even degree vertices
all even degree vertices
P: a path from v to v1
P': a portion of the path P that are in G'
e1
e2
e
Graph (5A) 29 Young Won Lim4/19/18
Euler Cycle – a path from v to v1
v1
v2
v3
v1
v2
v3
v1
v2
v3
v1
v2
v3
v1
v2
v3
v1
v2
v3
P P P
P ' P ' P '
Johnsonbough, Discrete Mathematics
Case 1: P ends at v1 Case 2: P ends at v
2 Case 3: P ends at v
3
v v v
v v v
Graph (5A) 30 Young Won Lim4/19/18
Euler Cycle
en.wikipedia.org
K
J
G
H
F
B
E
D
A
C I
ABCDEFGHIJK
Graph (5A) 31 Young Won Lim4/19/18
Euler Cycle
2 components
G G'
v1
v2
v3 v
1
v2
v3
Graph (5A) 32 Young Won Lim4/19/18
Euler Cycle
1 component
G G'
v1
v2
v3 v
1
v2
v3
Graph (5A) 33 Young Won Lim4/19/18
Euler Cycle
2 components
G G'
v1
v2
v3 v
1
v2
v3
Graph (5A) 34 Young Won Lim4/19/18
Euler Cycle
1 component
G G'
v1
v2
v3 v
1
v2
v3
Young Won Lim4/19/18
References
[1] http://en.wikipedia.org/[2]