Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Q77 a graph is planar if it can be drawn with no edges crossing. We treat the space outside of the graph as a face, called the outer face. For a connected planar graph G, we can identify a new graph, the planar dual G constructed as follows. The set of vertices of G is the set of faces of G. An edge in G exists when two faces share a common edge in G. So we can reuse the edge labels from G as edge labels in G. You can assume the result, that if G is connected, then G is also connected. Prove that, if G is connected, planar and bipartite, then G has an Eulerian circuit. Hint 1: all cycles in a bipartite graph are of even length. Hint 2: the face-degree of a face in G is its vertex-degree in G. Here is an example of a planar dual. Note that G is not bipartite. Using the graph G in figure 4 where G = ({A,..., E}, {e1,..., e7}) e2 B A €3 €6 e5 e1 ет e4 D E Figure 4: Example planar graph We construct the planar dual G* = ({w,...,2}, {e1,..., e7}) shown in figure 5. y e2 es A W " e1 e1 e4 D B €6 es Z e5 €7 X E e7 Figure 5: Graph G and it's planar dual G*

Q77 a graph is planar if it can be drawn with no edges crossing. We treat the space outside of the graph as a face, called the outer face. For a connected planar graph G, we can identify a new graph, the planar dual G constructed as follows. The set of vertices of G is the set of faces of G. An edge in G exists when two faces share a common edge in G. So we can reuse the edge labels from G as edge labels in G. You can assume the result, that if G is connected, then G is also connected. Prove that, if G is connected, planar and bipartite, then G has an Eulerian circuit. Hint 1: all cycles in a bipartite graph are of even length. Hint 2: the face-degree of a face in G is its vertex-degree in G. Here is an example of a planar dual. Note that G is not bipartite. Using the graph G in figure 4 where G = ({A,..., E}, {e1,..., e7}) e2 B A €3 €6 e5 e1 ет e4 D E Figure 4: Example planar graph We construct the planar dual G* = ({w,...,2}, {e1,..., e7}) shown in figure 5. y e2 es A W " e1 e1 e4 D B €6 es Z e5 €7 X E e7 Figure 5: Graph G and it's planar dual G*

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter12: Angle Relationships And Transformations
Section12.5: Reflections And Symmetry
Problem 20E
See similar textbooks
icon
Related questions
Question

Please do the following questions with handwritten working out

Q77 a graph is planar if it can be drawn with no edges crossing. We treat the space outside of the graph as a face, called the outer face. For a connected planar graph G, we can identify a new graph, the planar dual G constructed as follows. The set of vertices of G is the set of faces of G. An edge in G exists when two faces share a common edge in G. So we can reuse the edge labels from G as edge labels in G. You can assume the result, that if G is connected, then G is also connected. Prove that, if G is connected, planar and bipartite, then G has an Eulerian circuit. Hint 1: all cycles in a bipartite graph are of even length. Hint 2: the face-degree of a face in G is its vertex-degree in G. Here is an example of a planar dual. Note that G is not bipartite. Using the graph G in figure 4 where G = ({A,..., E}, {e1,..., e7}) e2 B A €3 €6 e5 e1 ет e4 D E Figure 4: Example planar graph We construct the planar dual G* = ({w,...,2}, {e1,..., e7}) shown in figure 5. y e2 es A W " e1 e1 e4 D B €6 es Z e5 €7 X E e7 Figure 5: Graph G and it's planar dual G*
Transcribed Image Text:Q77 a graph is planar if it can be drawn with no edges crossing. We treat the space outside of the graph as a face, called the outer face. For a connected planar graph G, we can identify a new graph, the planar dual G constructed as follows. The set of vertices of G is the set of faces of G. An edge in G exists when two faces share a common edge in G. So we can reuse the edge labels from G as edge labels in G. You can assume the result, that if G is connected, then G is also connected. Prove that, if G is connected, planar and bipartite, then G has an Eulerian circuit. Hint 1: all cycles in a bipartite graph are of even length. Hint 2: the face-degree of a face in G is its vertex-degree in G. Here is an example of a planar dual. Note that G is not bipartite. Using the graph G in figure 4 where G = ({A,..., E}, {e1,..., e7}) e2 B A €3 €6 e5 e1 ет e4 D E Figure 4: Example planar graph We construct the planar dual G* = ({w,...,2}, {e1,..., e7}) shown in figure 5. y e2 es A W " e1 e1 e4 D B €6 es Z e5 €7 X E e7 Figure 5: Graph G and it's planar dual G*
Expert Solution
steps

Step by step

Solved in 2 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Holt Mcdougal Larson Pre-algebra: Student Edition…
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra
College Algebra
Algebra
ISBN:
9781337282291
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS