problems of graph theory · graph terminology i a simple graph has no loops or multiple edges. i a...
TRANSCRIPT
Problems of Graph Theory
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Overview
Path Problems
Isomorphic Graphs
Degree Sequences
Shortest Paths
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Outline
Path Problems
Isomorphic Graphs
Degree Sequences
Shortest Paths
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The Konigsberg Bridges
Can one find a route that would cross each of the sevenbridges exactly once?
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Euler’s Approach
The graph on the vertices A,B ,C ,D with edgesa, b, c , d , e, f , g does not have an Euler trail.
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Graph Terminology
I A simple graph has no loops or multiple edges.
I A walk is a sequence of edges, each consecutive pair ofwhich is incident to a common vertex.
I A closed walk is one in which the initial and final verticescoincide.
I A trail is a walk in which all edges are distinct.
I A circuit is a closed trail.
I A path is a trail in which all vertices distinct (with thepossible exception of the initial and final verticescoinciding).
I A cycle is a closed path.
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Euler Trails
Determine whether the graph has an Euler trail (i.e., a trailwhich contains every edge of the graph).
b c d
a
g
e f
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Hamilton PathsDetermine whether the graph has a Hamilton path (i.e., apath which contains every vertex of the graph).
a b
c
d
e
f
g
h
i
j k
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Hamilton Cycles
Determine whether each graph has a Hamilton cycle (i.e., acycle which contains every vertex of the graph).
a bc
d
e
f
g
h
ij k
a
b
c
d
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f
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h
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Outline
Path Problems
Isomorphic Graphs
Degree Sequences
Shortest Paths
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Isomorphic Graphs
DefinitionTwo graphs are isomorphic if they are essentially the same,even though they may appear different, in the sense that onecan be transformed into the other by moving vertices aroundand changing the lengths of some of the edges that join them.
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Example
Determine whether the following graphs are isomorphic.
a
b
d
c
a
b
c
d
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Example
Determine whether the following graphs are isomorphic.
a
b
c
d
f e
a
b
c
d
f e
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Example
Determine whether the following graphs are isomorphic.
a
b
c
d
e
a
b
c
d
e
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Outline
Path Problems
Isomorphic Graphs
Degree Sequences
Shortest Paths
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Definition of Degree Sequence
DefinitionThe degree sequence of a simple graph is a list, or sequence,of all of the degrees of the vertices in the graph including anyrepetitions.
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Example
The following graph has degree sequence (5, 5, 4, 4, 4, 4, 4).
b c d
a
g
e f
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Question
Which sequences are degree sequences of graphs?
Note that there is no simple graph having degree sequence(3, 2, 2). Why?
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Question
Which sequences are degree sequences of graphs?
Note that there is no simple graph having degree sequence(3, 2, 2). Why?
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Handshaking Lemma
LemmaThe sum of the degrees of the vertices of a graph is twice thenumber of edges.
CorollaryIn any graph, the sum of the vertex degrees is even.
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Example
Determine which of the following sequences are degreesequences for some simple graph.
(a) (5, 5, 5, 4, 4, 3, 3)
(b) (7, 6, 5, 4, 3, 2, 1)
(c) (6, 5, 4, 3, 2, 2, 1)
(d) (6, 5, 4, 3, 2, 1, 0)
(e) (3, 3, 2, 2, 2, 1, 1)
(f) (5, 5, 4, 4, 4, 3, 3)
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Existence of Vertices of Equal Degree
PropositionIn any simple graph having two or more vertices, there mustexist at least two vertices having the same degree.
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Outline
Path Problems
Isomorphic Graphs
Degree Sequences
Shortest Paths
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Shortest Path Problem
Find a shortest route between vertices in a given graph, whichmay be undirected, directed, weighted, etc.
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Undirected Graph
Find a shortest path (i.e., a path with fewest edges)from a to h.
a
b
c
d
e
f
g
h
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Directed Graph
Find a shortest path (i.e., a path with fewest edges) path froma to h, respecting the edge directions.
a
b
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d
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f
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h
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Acknowledgements
Statements of results follow the notation and wording ofAnderson’s First Course in Discrete Mathematics.
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