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Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets of vertices U,W whose union is V where any path of G from vertices of U to vertices of W contains e.

Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets of vertices U,W whose union is V where any path of G from vertices of U to vertices of W contains 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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**Graph Theory Problem:** Let \( G = (V, E) \) be a connected graph with a bridge \( e = uv \). Prove that there exist two disjoint sets of vertices \( U, W \) whose union is \( V \) where any path of \( G \) from vertices of \( U \) to vertices of \( W \) contains \( e \). **Analysis:** - **Graph and Bridge:** - \( G \) is a graph with vertices \( V \) and edges \( E \). - The edge \( e = uv \) is a bridge, meaning if it is removed, the graph becomes disconnected. - **Sets \( U \) and \( W \):** - You need to find two disjoint sets of vertices, \( U \) and \( W \), such that their union equals the entire vertex set \( V \). - Every path from a vertex in \( U \) to a vertex in \( W \) must include the bridge \( e \). **Proof Strategy:** 1. **Identify Subsets:** - Consider the graph \( G - e \), which indicates the graph \( G \) without the bridge \( e \). - In \( G - e \), the graph becomes disconnected into two components because \( e \) is a bridge. - Let one component of \( G - e \) be \( C_1 \) containing vertex \( u \), and the other be \( C_2 \) containing vertex \( v \). 2. **Define \( U \) and \( W \):** - Set \( U \) to be the set of all vertices in \( C_1 \). - Set \( W \) to be the set of all vertices in \( C_2 \). 3. **Check Conditions:** - Union: Since every vertex in \( G \) is in either \( C_1 \) or \( C_2 \), \( U \cup W = V \). - Disjoint: By definition, \( U \) and \( W \) are disjoint. - Path Condition: Any path from a vertex in \( U \) to a vertex in \( W \) in \( G \) must pass through edge \( e = uv \). **Conclusion:** The sets \( U \) and \( W \) satisfy the required conditions, thus proving
Transcribed Image Text:**Graph Theory Problem:** Let \( G = (V, E) \) be a connected graph with a bridge \( e = uv \). Prove that there exist two disjoint sets of vertices \( U, W \) whose union is \( V \) where any path of \( G \) from vertices of \( U \) to vertices of \( W \) contains \( e \). **Analysis:** - **Graph and Bridge:** - \( G \) is a graph with vertices \( V \) and edges \( E \). - The edge \( e = uv \) is a bridge, meaning if it is removed, the graph becomes disconnected. - **Sets \( U \) and \( W \):** - You need to find two disjoint sets of vertices, \( U \) and \( W \), such that their union equals the entire vertex set \( V \). - Every path from a vertex in \( U \) to a vertex in \( W \) must include the bridge \( e \). **Proof Strategy:** 1. **Identify Subsets:** - Consider the graph \( G - e \), which indicates the graph \( G \) without the bridge \( e \). - In \( G - e \), the graph becomes disconnected into two components because \( e \) is a bridge. - Let one component of \( G - e \) be \( C_1 \) containing vertex \( u \), and the other be \( C_2 \) containing vertex \( v \). 2. **Define \( U \) and \( W \):** - Set \( U \) to be the set of all vertices in \( C_1 \). - Set \( W \) to be the set of all vertices in \( C_2 \). 3. **Check Conditions:** - Union: Since every vertex in \( G \) is in either \( C_1 \) or \( C_2 \), \( U \cup W = V \). - Disjoint: By definition, \( U \) and \( W \) are disjoint. - Path Condition: Any path from a vertex in \( U \) to a vertex in \( W \) in \( G \) must pass through edge \( e = uv \). **Conclusion:** The sets \( U \) and \( W \) satisfy the required conditions, thus proving
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