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Young Won Lim5/11/18
Graph Overview (1A)
Young Won Lim5/11/18
Copyright (c) 2015 – 2018 Young W. Lim.
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Graph Overview (1A) 3 Young Won Lim5/11/18
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 Overview (1A) 4 Young Won Lim5/11/18
Complete Graphs
https://en.wikipedia.org/wiki/Complete_graph
Graph Overview (1A) 5 Young Won Lim5/11/18
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 is connected.
Graph Overview (1A) 6 Young Won Lim5/11/18
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 Overview (1A) 7 Young Won Lim5/11/18
Complete Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph Overview (1A) 8 Young Won Lim5/11/18
Complete Bipartite Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
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Star Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph Overview (1A) 10 Young Won Lim5/11/18
Wheel Graphs
https://en.wikipedia.org/wiki/Gallery_of_named_graphs
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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)
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Planar Graphs
https://en.wikipedia.org/wiki/Planar_graph
A planar graph and its dual
Graph Overview (1A) 13 Young Won Lim5/11/18
Cycle Graphs
https://en.wikipedia.org/wiki/Cycle_graphhttps://en.wikipedia.org/wiki/Gallery_of_named_graphs
Graph Overview (1A) 14 Young Won Lim5/11/18
Tree Graphs
https://en.wikipedia.org/wiki/Cycle_graph
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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
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Gray Code
https://en.wikipedia.org/wiki/Gray_code
Tesseract
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Adjacency Lists
https://en.wikipedia.org/wiki/Adjacency_list
Graph Overview (1A) 18 Young Won Lim5/11/18
Incidence Matrix
https://en.wikipedia.org/wiki/Incidence_matrix
Graph Overview (1A) 19 Young Won Lim5/11/18
Adjacency Matrix
https://en.wikipedia.org/wiki/Adjacency_matrix
Graph Overview (1A) 20 Young Won Lim5/11/18
Hamiltonian Path
https://en.wikipedia.org/wiki/Path_(graph_theory)
Graph Overview (1A) 21 Young Won Lim5/11/18
Minimum Spanning Tree
https://en.wikipedia.org/wiki/Minimum_spanning_tree
Graph Overview (1A) 22 Young Won Lim5/11/18
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 Overview (1A) 23 Young Won Lim5/11/18
Shortest path problem
https://en.wikipedia.org/wiki/Shortest_path_problem
Graph Overview (1A) 24 Young Won Lim5/11/18
Traveling salesman problem
https://en.wikipedia.org/wiki/Travelling_salesman_problem
Graph Overview (1A) 25 Young Won Lim5/11/18
Simple Graph
https://en.wikipedia.org/wiki/Travelling_salesman_problem
A simple graph is an undirected graph without multiple edges or loops.
the edges form a set (rather than a multiset)each edge is an unordered pair of distinct vertices.
can define a simple graph to be a set V of vertices together with a set E of edges,
E are 2-element subsets of V
with n vertices, the degree of every vertex is at most n − 1
K
J
G
H
F
B
E
D
A
C I
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Multi-Graph
https://en.wikipedia.org/wiki/Travelling_salesman_problem
A multigraph, as opposed to a simple graph, is an undirected graph in which multiple edges (and sometimes loops) are allowed.
B
E D
A
C
B
E
DA C
F
A B C
D
E FG
H
Graph Overview (1A) 27 Young Won Lim5/11/18
Multiple Edges
https://en.wikipedia.org/wiki/Travelling_salesman_problem
● multiple edges ● parallel edges● Multi-edges
are two or more edges that are incident to the same two vertices
A simple graph has no multiple edges.
Graph Overview (1A) 28 Young Won Lim5/11/18
Loop
https://en.wikipedia.org/wiki/Travelling_salesman_problem
● a loop● a self-loop● a buckle
is an edge that connects a vertex to itself.
A simple graph contains no loops.
Graph Overview (1A) 29 Young Won Lim5/11/18
Walks
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
For a graph G= (V, E), a walk is defined as a sequence
of alternating vertices and edges such as
where each edge
The length of this walk is
Edges are allowed to be repeated
v0, e1, v1, e2, ⋯ , ek , vk
ei = {vi−1 , v i}
kv0
e1
v1
e2
v2
e3
v3
ek
vk
⋯
B
E D
A
CABCDE
ABCDCBE
e i=e j for some i , j
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Open / Closed Walks
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
A walk is considered to be closed if the starting vertex is the same as the ending vertex.
Otherwise open
v0
e1
v1
e2
v2
e3
v3
ek
vk = v0
⋯
B
E D
A
C
ABCDE
ABCDCBE
ABCDA
Graph Overview (1A) 31 Young Won Lim5/11/18
Open / Closed Walks
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
A walk is considered to be closed if the starting vertex is the same as the ending vertex.
Otherwise open
v0
e1
v1
e2
v2
e3
v3
ek
vk = v0
⋯
B
E D
A
C
ABCDE
ABCDCBE
ABCDAclosed walk
open walk
open walk
Graph Overview (1A) 32 Young Won Lim5/11/18
Trails
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
A trail is defined as a walk with no repeated edges.
B
E D
A
C
ABCDE
ABCDCBE
ABCDAclosed walk
open walk
open walk
v0
e1
v1
e2
v2
e3
v3
ek
vk
⋯
e i ≠ e j for all i , j
closed trail
open trail
open trail
Graph Overview (1A) 33 Young Won Lim5/11/18
Paths
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
A path is defined as a open trail with no repeated vertices.
B
E D
A
C
ABCDE
ABCDCBE
ABCDAclosed walk
open walk
open walk
v0
e1
v1
e2
v2
e3
v3
ek
vk ≠ v0
⋯
e i ≠ e j for all i , j
closed trail
open trail
open trail
v i ≠ v j for all i , j
path
path
path
BEDABCopen walkopen trailpath
Graph Overview (1A) 34 Young Won Lim5/11/18
Cycles
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
A cycle is defined as a closed trail with no repeated vertices except the start/end vertex
B
E D
A
C
ABCDAclosed walk
v0
e1
v1
e2
v2
e3
v3
ek
vk = v0
⋯
e i ≠ e j for all i , j
circuit
v i ≠ v j for all i , j
cycle
ABCDEBDAclosed walkcircuitcycle
Graph Overview (1A) 35 Young Won Lim5/11/18
Circuits
http://mathonline.wikidot.com/walks-trails-paths-cycles-and-circuits
A circuit is defined as a closed trail with possibly repeated vertices but with no repeated edges
v0
e1
v1
e2
v2
e3
v3
ek
vk = v0
⋯
e i ≠ e j for all i , j
v i = v j for some i , j
B
E D
A
C
ABCDAclosed walkcircuit
ABCDEBDAclosed walkcircuit
Graph Overview (1A) 36 Young Won Lim5/11/18
Walk, Trail, Path, Circuit, Cycle
open walks closed walks
trails circuits
path cycle
e i ≠ e j e i ≠ e j
v i ≠ v j v i ≠ v j
v0 ≠ vk v0 = vk
Graph Overview (1A) 37 Young Won Lim5/11/18
Walk, Trail, Path, Circuit, Cycle
Vertices Edges
Walk may may (Closed/Open) repeat repeat
Trail may cannot (Open) repeat repeat
Path cannot cannot (Open) repeat repeat
Circuit may cannot (Closed) repeat repeat
Cycle cannot cannot (Closed) repeat repeat
https://math.stackexchange.com/questions/655589/what-is-difference-between-cycle-path-and-circuit-in-graph-theory
Young Won Lim5/11/18
References
[1] http://en.wikipedia.org/[2]