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Magazine Chapter 10: Problem 52
Show that if \(G\) is a connected graph with \(n\) vertices then a) \(\kappa(G)=n-1\) if and only if \(G=K_{n}\) b) \(\lambda(G)=n-1\) if and only if \(G=K_{n}\)
Short Answer
Expert verified
a) \kappa(G)=n-1\ if and only if \G=K_{n}\. b) \lambda(G)=n-1\ if and only if \G=K_{n}\.
Step by step solution
01
Understanding the Problem
Given a connected graph G with n vertices, need to prove two statements about its connectivity and edge-connectivity.
02
Define Terminologies
Recall that \(\backslashkappa(G)\) represents vertex connectivity and \(\backslashlambda(G)\) represents edge connectivity. \K_{n}\ is the complete graph with n vertices, meaning every pair of vertices is connected by an edge.
03
Step 1a: Prove \(\backslashkappa(G)=n-1\) if and only if \G=K_{n}\
Show that for any connected graph \G\ with n vertices, \(\backslashkappa(G)=n-1\) only holds if every subset of \(n-1\) vertices is connected. This is true if and only if G is a complete graph \(K_{n}\).
04
Step 1a: Forward Direction - If \(G=K_{n}\), then \(\backslashkappa(G)=n-1\)
In complete graph \(K_{n}\), removing up to \(n-2\) vertices leaves a connected subgraph with at least 2 vertices connected. Removing \(n-1\) vertices leaves a single vertex, thus the minimum number needed to disconnect \(n-1\). Hence, \(\backslashkappa(K_{n})=n-1\).
05
Step 1a: Reverse Direction - If \(\backslashkappa(G)=n-1\), then G is \(K_{n}\)
If \(G\) is not complete, a vertex belongs to less than \(n-1\) vertices. By removing these vertices, G becomes disconnected. Therefore, to satisfy \(\backslashkappa(G)=n-1\), \(G\) must be \(K_{n}\).
06
Step 1b: Prove \(\backslashlambda(G)=n-1\) if and only if \G=K_{n}\
Show that for any connected graph, the edge connectivity \(\backslashlambda(G)=n-1\) only holds if G is \(K_{n}\), meaning removing \(n-1\) edges can disconnect G.
07
Step 1b: Forward Direction - If \(G=K_{n}\), then \(\backslashlambda(G)=n-1\)
In \(K_{n}\), every vertex is connected to \(n-1\) edges. Removing any \(n-1\) edges leaves vertices still connected by remaining edges. Therefore, removing \(n-1\) edges is minimal for disconnection. Thus, \(\backslashlambda(K_{n})=n-1\).
08
Step 1b: Reverse Direction - If \(\backslashlambda(G)=n-1\), then G is \(K_{n}\)
If \(G\) is not complete and less than \(n-1\) edges are present, removing these edges will sufficiently disconnect G. To satisfy \(\backslashlambda(G)=n-1\), G must be \(K_{n}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertex Connectivity
Vertex connectivity, denoted as \(\backslashkappa(G)\), is a measure of a graph's robustness related to its vertices. Specifically, \(\backslashkappa(G)\) counts the smallest number of vertices that need to be removed to disconnect the graph or make it trivial (reduce it to a single vertex). For instance, in a simple and connected graph, if removing two vertices makes the graph disconnected, then the vertex connectivity is 2. This concept helps us understand how well-connected the graph is in terms of nodes.
For example, if we have a complete graph \(K_n\), all vertices are interconnected. Removing up to \(n-2\) vertices still keeps the graph connected. The only way to disconnect it is by removing \(n-1\) vertices. Hence, for a complete graph with \(n\) vertices, we conclude that \(\backslashkappa(K_n) = n-1\).
On the other hand, if a graph \(G\) has a vertex connectivity of \(n-1\), removing \((n-1)\) vertices would leave a single vertex isolated. This can only happen if every possible pair of vertices was initially connected, i.e., the graph is complete. Thus, \(G = K_n\).
For example, if we have a complete graph \(K_n\), all vertices are interconnected. Removing up to \(n-2\) vertices still keeps the graph connected. The only way to disconnect it is by removing \(n-1\) vertices. Hence, for a complete graph with \(n\) vertices, we conclude that \(\backslashkappa(K_n) = n-1\).
On the other hand, if a graph \(G\) has a vertex connectivity of \(n-1\), removing \((n-1)\) vertices would leave a single vertex isolated. This can only happen if every possible pair of vertices was initially connected, i.e., the graph is complete. Thus, \(G = K_n\).
Edge Connectivity
Edge connectivity, denoted as \(\backslashlambda(G)\), refers to the smallest number of edges that need to be removed to disconnect the graph. This is an important measure of a graph's resilience regarding its edges. For instance, if removing three edges makes a graph disconnected, its edge connectivity is 3.
Consider a complete graph \(K_n\). Each vertex connects to every other vertex, meaning each vertex has \((n-1)\) edges. Even if we remove \((n-1)\) edges, there are still enough connections to keep the graph together. Therefore, the minimum number of edges that need to be removed to cause a disconnection in \(K_n\) is \((n-1)\), so \(\backslashlambda(K_n) = n-1\).
If a given graph \(G\) has edge connectivity \((n-1)\), removing \((n-1)\) edges would make it disconnected. This implies that initially, every vertex was connected to \((n-1)\) other vertices, which is a characteristic of a complete graph. Hence, \(G\) must be \(K_n\).
Consider a complete graph \(K_n\). Each vertex connects to every other vertex, meaning each vertex has \((n-1)\) edges. Even if we remove \((n-1)\) edges, there are still enough connections to keep the graph together. Therefore, the minimum number of edges that need to be removed to cause a disconnection in \(K_n\) is \((n-1)\), so \(\backslashlambda(K_n) = n-1\).
If a given graph \(G\) has edge connectivity \((n-1)\), removing \((n-1)\) edges would make it disconnected. This implies that initially, every vertex was connected to \((n-1)\) other vertices, which is a characteristic of a complete graph. Hence, \(G\) must be \(K_n\).
Complete Graph
A complete graph, denoted as \K_n\, is a type of graph where every pair of distinct vertices is connected by a unique edge. This is the most interconnected kind of graph. For \(n\) vertices in a complete graph, there are \(\frac{n(n-1)}{2}\) edges, since each vertex is joined with every other vertex.
The beauty of a complete graph lies in its maximum connectivity properties:
The beauty of a complete graph lies in its maximum connectivity properties:
- Vertex Connectivity: Removing \((n-1)\) vertices disconnects or trivializes the graph, meaning \(\backslashkappa(K_n) = n-1\).
- Edge Connectivity: Removing \((n-1)\) edges still leaves the graph connected, thus \(\backslashlambda(K_n) = n-1\).
