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9.6 Prove or disprove the following.

9.6 Prove or disprove the following.

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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## 9.6 Prove or Disprove the Following Statements (a) **Every subgraph of a planar graph is planar.** (b) **Every subgraph of a nonplanar graph is nonplanar.** (c) **If G is a nonplanar graph, then G contains a proper nonplanar subgraph.** (d) **If G does not contain \( K_5 \) or \( K_{3,3} \) as a subgraph, then G is planar.** **Explanation of Terms:** - A **planar graph** is a graph that can be embedded in the plane, meaning it can be drawn on a plane without any edges crossing. - A **subgraph** is a graph formed from a subset of the vertices and edges of another graph. - \( K_5 \) is the complete graph on 5 vertices, and \( K_{3,3} \) is the complete bipartite graph with partitions of 3 vertices. These are known for their properties related to planarity (Kuratowski's theorem).
Transcribed Image Text:## 9.6 Prove or Disprove the Following Statements (a) **Every subgraph of a planar graph is planar.** (b) **Every subgraph of a nonplanar graph is nonplanar.** (c) **If G is a nonplanar graph, then G contains a proper nonplanar subgraph.** (d) **If G does not contain \( K_5 \) or \( K_{3,3} \) as a subgraph, then G is planar.** **Explanation of Terms:** - A **planar graph** is a graph that can be embedded in the plane, meaning it can be drawn on a plane without any edges crossing. - A **subgraph** is a graph formed from a subset of the vertices and edges of another graph. - \( K_5 \) is the complete graph on 5 vertices, and \( K_{3,3} \) is the complete bipartite graph with partitions of 3 vertices. These are known for their properties related to planarity (Kuratowski's theorem).
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