Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN: 9780133594140
Author: James Kurose, Keith Ross
Publisher: PEARSON
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**Problem 1:** Determine whether graphs \( G \) and \( H \) are planar or not. To show planarity, give a planar embedding. To show that a graph is not planar, use Kuratowski’s theorem.

### Graph G:
Graph \( G \) consists of seven vertices labeled \( a, b, c, d, e, f, \) and \( g \). The vertices are interconnected with edges forming a complete graph structure, indicating a high degree of connectivity among them.

### Graph H:
Graph \( H \) also has seven vertices labeled \( a, b, c, d, e, f, \) and \( g \). This graph appears more interconnected and complex, with multiple crisscrossing edges, indicating possible non-planarity.

**Explanation of Planarity and Kuratowski's Theorem:**

- **Planarity**: A graph is planar if it can be drawn on a plane without any edges crossing.
- **Kuratowski's Theorem**: A graph is non-planar if and only if it contains a subgraph that is a subdivision of \( K_5 \) (complete graph on five vertices) or \( K_{3,3} \) (complete bipartite graph on two sets of three vertices).

To determine the planarity of Graphs \( G \) and \( H \), examine whether these graphs can be redrawn without crossing edges or if they contain a subgraph homeomorphic to \( K_5 \) or \( K_{3,3} \).
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Transcribed Image Text:**Problem 1:** Determine whether graphs \( G \) and \( H \) are planar or not. To show planarity, give a planar embedding. To show that a graph is not planar, use Kuratowski’s theorem. ### Graph G: Graph \( G \) consists of seven vertices labeled \( a, b, c, d, e, f, \) and \( g \). The vertices are interconnected with edges forming a complete graph structure, indicating a high degree of connectivity among them. ### Graph H: Graph \( H \) also has seven vertices labeled \( a, b, c, d, e, f, \) and \( g \). This graph appears more interconnected and complex, with multiple crisscrossing edges, indicating possible non-planarity. **Explanation of Planarity and Kuratowski's Theorem:** - **Planarity**: A graph is planar if it can be drawn on a plane without any edges crossing. - **Kuratowski's Theorem**: A graph is non-planar if and only if it contains a subgraph that is a subdivision of \( K_5 \) (complete graph on five vertices) or \( K_{3,3} \) (complete bipartite graph on two sets of three vertices). To determine the planarity of Graphs \( G \) and \( H \), examine whether these graphs can be redrawn without crossing edges or if they contain a subgraph homeomorphic to \( K_5 \) or \( K_{3,3} \).
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