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how that each of the following graphs are planar by providing a planar representation of the graph. e sure to label the vertices! (There is a good, free, online software program called Geogebra that you an use to help with this) (a) (b) (c) H F E G B E 0 E

how that each of the following graphs are planar by providing a planar representation of the graph. e sure to label the vertices! (There is a good, free, online software program called Geogebra that you an use to help with this) (a) (b) (c) H F E G B E 0 E

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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1. **Show that each of the following graphs are planar by providing a planar representation of the graph. Be sure to label the vertices! (There is a good, free, online software program called Geogebra that you can use to help with this)** (a) The first graph consists of vertices labeled \( A, B, C, D, E, F, G \). All vertices are connected by various edges, including: - Connections from \( A \) to \( B, C, D, E, F \) - \( B \) connects to \( F, G \) - \( D \) connects to \( C \) - There is a zigzag pattern of connections from the top vertex \( C \) down to \( G \). (b) The second graph is an octagon with vertices labeled \( A, B, C, D, E, F, G, H \). - Each vertex on the perimeter is connected to its adjacent vertices creating an outer ring. - Cross connections exist significantly within the shape: - \( A \) directly connects to \( C, E \) - \( B \) directly connects to \( D, F \) - \( D \) connects to \( F \) - Overall, creating a web of triangular connections within. (c) The third graph forms a star with vertices labeled \( A, B, C, D, E, F, G, H \). - The top point of the star is \( A \) connecting to \( B, C \). - The five outer points \( B, C, D, E, F \) create star-shaped outer connections, enclosing a pentagon. - The inside is criss-crossed with connections forming a smaller central pentagon formed by points \( B, C, D, E, F \) and \( H \). These diagrams are used to illustrate different configurations and help determine if a given graph is planar, which means it can be drawn on a plane without any of its edges crossing, except at their endpoints.
Transcribed Image Text:1. **Show that each of the following graphs are planar by providing a planar representation of the graph. Be sure to label the vertices! (There is a good, free, online software program called Geogebra that you can use to help with this)** (a) The first graph consists of vertices labeled \( A, B, C, D, E, F, G \). All vertices are connected by various edges, including: - Connections from \( A \) to \( B, C, D, E, F \) - \( B \) connects to \( F, G \) - \( D \) connects to \( C \) - There is a zigzag pattern of connections from the top vertex \( C \) down to \( G \). (b) The second graph is an octagon with vertices labeled \( A, B, C, D, E, F, G, H \). - Each vertex on the perimeter is connected to its adjacent vertices creating an outer ring. - Cross connections exist significantly within the shape: - \( A \) directly connects to \( C, E \) - \( B \) directly connects to \( D, F \) - \( D \) connects to \( F \) - Overall, creating a web of triangular connections within. (c) The third graph forms a star with vertices labeled \( A, B, C, D, E, F, G, H \). - The top point of the star is \( A \) connecting to \( B, C \). - The five outer points \( B, C, D, E, F \) create star-shaped outer connections, enclosing a pentagon. - The inside is criss-crossed with connections forming a smaller central pentagon formed by points \( B, C, D, E, F \) and \( H \). These diagrams are used to illustrate different configurations and help determine if a given graph is planar, which means it can be drawn on a plane without any of its edges crossing, except at their endpoints.
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