Graph
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Lecture 18-19
Euler Graph
An Euler circuit in a graph G is a simple circuit containing every edge of G.
A connected graph contains an Eulerian Circuit if and only if every vertex has even degree.
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only if every vertex has even degree.
An Euler path in G is a simple path containing every edge of G.
A connected graph contains an Eulerian Path if and only if exactly 2 vertices have odd degree.
Euler Graph
Which of the undirected graphs in Figure 3 have an Euler circuit?
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G1 : ?
Euler Graph
Which of the undirected graphs in Figure 3 have an Euler circuit?
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G1 : Yes, a, e, c, d, e, b, a.G2 : ?
Euler Graph
Which of the undirected graphs in Figure 3 have an Euler circuit?
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G1 : Yes, a, e, c, d, e, b, a.G2 : NoG3 : ?
Euler GraphWhich of the undirected graphs in Figure 3 have an Euler circuit?
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G1 : Yes, a, e, c, d, e, b, a.G2 : NoG3 : No
Euler Graph
Which of the undirected graphs in Figure 3 have an Euler path?
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G2 : ?
Euler Graph
Which of the undirected graphs in Figure 3 have an Euler path?
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G2 : NoG3: ?
Euler Graph
Which of the undirected graphs in Figure 3 have an Euler path?
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G2 : NoG3 : Yes: a, c, d, e, b, d, a, b.
Hamiltonian Cycles
An Euler cycle is a cycle in a graph that includes each edge exactly once. Examples: Designing and optimizing routes refuse trucks, snow ploughs, or postmen. In all of these applications, every edge in a graph must be traversed at least once.
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traversed at least once. A Hamiltonian cycle is a cycle in a graph G that
contains each vertex exactly once except for the starting and ending vertex that appears twice. Examples: travelling sales man who wishes to visit every city and also minimize his route to each city and return to his home city.
Hamilton Circuit
A Hamilton path of a graph or digraph is a path that contains each vertex exactly once, except that the end vertices may be the same.
A Hamilton circuit (or cycle) is a Hamilton path that is a
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A Hamilton circuit (or cycle) is a Hamilton path that is a cycle.
Contrast this with an Euler circuit which contains each edge exactly once.
Hamilton Path and Circuit
G has Hamilton path but no Hamilton circuit (or cycle).
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ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Hamilton Circuit Find a Hamilton circuit in the G.
b
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ba c
d e f g h
i j
Solution: d,a
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g,c
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g,c,h
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g,c,h,j
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g,c,h,j,f
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g,c,h,j,f,i
ba c
Hamilton Circuit Find a Hamilton circuit in the G.
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a c
d e f g h
i j
Solution: d,a,e,b,g,c,h,j,f,i,d
ba c
Hamilton Circuit Find another Hamilton circuit in the same graph G.
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a c
d e f g h
i j
Note that while verticies a and b were on previous Hamilton circuit the edge was not.
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
ba c
Hamilton Circuit Find another Hamilton circuit in the G.
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a c
d e f g h
i j
Euler and Hamilton cycles
A
B
C
D
F
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G has a Euler cycle but no Hamilton cycle
B E
Hamilton cycles
Planar Graphs
Planar graphs are graphs that can be drawn in the plane without edges having to cross.
Understanding planar graph is important:
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Any graph representation of maps/ topographical information is planar. graph algorithms often specialized to planar
graphs (e.g. traveling salesperson) Circuits usually represented by planar
graphs
Planar Graphs-Common Misunderstanding
Just because a graph is drawn with edges crossing doesnt mean its not planar.
Q: Why cant we conclude that the following is non-planar?
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Planar Graphs-Common Misunderstanding
A: Because it is isomorphic to a graph which is planar:
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Thank You
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