Lecture 18-19 Graph Tree

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


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G1 : Yes, a, e, c, d, e, b, a.G2 : ?

Euler Graph


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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


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G2 : NoG3: ?

Euler Graph


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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


b

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ba c

d e f g h

i j

Solution: d,a

ba c


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a c

d e f g h

i j

Solution: d,a,e

ba c


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a c

d e f g h

i j

Solution: d,a,e,b

ba c


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a c

d e f g h

i j

Solution: d,a,e,b,g

ba c


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a c

d e f g h

i j

Solution: d,a,e,b,g,c

ba c


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a c

d e f g h

i j

Solution: d,a,e,b,g,c,h

ba c


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a c

d e f g h

i j

Solution: d,a,e,b,g,c,h,j

ba c


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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


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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


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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


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a c

d e f g h

i j

ba c


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a c

d e f g h

i j

ba c


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a c

d e f g h

i j

ba c


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a c

d e f g h

i j

ba c


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a c

d e f g h

i j

ba c


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a c

d e f g h

i j

ba c


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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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Lecture 18-19 Graph Tree