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How many faces does the below planar graph contains?

How many faces does the below planar graph contains?

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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**Understanding Planar Graph Faces** In graph theory, a planar graph is a graph that can be embedded in the plane, meaning it can be drawn on a plane in such a way that its edges intersect only at their endpoints. When examining a planar graph, one interesting feature to count is the number of faces it contains. A face in a planar graph is a region bounded by edges, including the outer infinite region. To help explain this, let's analyze the following question and corresponding diagram: **Question:** *How many faces does the below planar graph contain?* **Graph Description:** The provided image displays a planar graph consisting of 7 vertices and 11 edges, arranged in a somewhat irregular manner. The vertices (represented as black dots) are interconnected with edges (straight lines), which form multiple enclosed areas (faces). **Counting the Faces:** Here's a step-by-step approach to count the faces in the given planar graph: 1. **Identify each enclosed region:** Each set of interconnected edges forms a boundary of a face. Trace each edge and notice enclosed regions. 2. **Remember the outer face:** The area surrounding the graph also counts as a face. **Diagram Explanation:** This specific diagram forms 6 enclosed regions, plus the outer infinite region, making a total of 7 faces. To summarize, the graph contains 7 faces, including the outside region. Understanding and identifying the faces of planar graphs are essential skills in graph theory, which find applications in various fields such as computer networks, circuit design, and geographical mapping.
Transcribed Image Text:**Understanding Planar Graph Faces** In graph theory, a planar graph is a graph that can be embedded in the plane, meaning it can be drawn on a plane in such a way that its edges intersect only at their endpoints. When examining a planar graph, one interesting feature to count is the number of faces it contains. A face in a planar graph is a region bounded by edges, including the outer infinite region. To help explain this, let's analyze the following question and corresponding diagram: **Question:** *How many faces does the below planar graph contain?* **Graph Description:** The provided image displays a planar graph consisting of 7 vertices and 11 edges, arranged in a somewhat irregular manner. The vertices (represented as black dots) are interconnected with edges (straight lines), which form multiple enclosed areas (faces). **Counting the Faces:** Here's a step-by-step approach to count the faces in the given planar graph: 1. **Identify each enclosed region:** Each set of interconnected edges forms a boundary of a face. Trace each edge and notice enclosed regions. 2. **Remember the outer face:** The area surrounding the graph also counts as a face. **Diagram Explanation:** This specific diagram forms 6 enclosed regions, plus the outer infinite region, making a total of 7 faces. To summarize, the graph contains 7 faces, including the outside region. Understanding and identifying the faces of planar graphs are essential skills in graph theory, which find applications in various fields such as computer networks, circuit design, and geographical mapping.
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