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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
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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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*
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