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*
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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*
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