- #1
Chinta
- 8
- 0
The problem is as follows:
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Let G be a connected graph.
For a vertex x of G we denote by G−x the graph formed by removing x and all edges incident on x from G. G is said to be good if there are at
least two distinct vertices x,y in G such that both G − x and G − y are connected.
(i) Show that for any subgraph H of G, H is good if and only if G is good.
(ii) Given a good graph, devise a linear time algorithm to find two such vertices.
(iii) Show that there exists a graph G such that we cannot find three distinct vertices
u1,u2,u3such that each of G − u1,G − u2, and G − u3 is connected.
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I have approached to some extent. Here is what I've done:
If G is a connected graph then we can easily find its spanning tree and then removing any two leaves x and y from the tree one at a time gives us 2 another graphs G − x and G − y such that they are both connected.
"Hence any connected graph G is a good graph."
1. I'm stuck on how to solve part (i).
2. I have designed the algorithm for the part(ii) which extracts the spanning tree from G and then finds two such leaves x and y. Clearly
it is linear in terms of #vertices(=n) & #edges(=m) i.e. O(n+m)
3. I have also solved the third part with a graph as follows:
O------------------O-----------------O
In the above graph G we denote three vertices as u1, u2, u3. The edges are (u1,u2) and (u2,u3). Clearly G − u2 is not connected.
Please help me in solving part(i).
Thanks in advance :)
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Let G be a connected graph.
For a vertex x of G we denote by G−x the graph formed by removing x and all edges incident on x from G. G is said to be good if there are at
least two distinct vertices x,y in G such that both G − x and G − y are connected.
(i) Show that for any subgraph H of G, H is good if and only if G is good.
(ii) Given a good graph, devise a linear time algorithm to find two such vertices.
(iii) Show that there exists a graph G such that we cannot find three distinct vertices
u1,u2,u3such that each of G − u1,G − u2, and G − u3 is connected.
---------------------------------------------------------------------------------------------------------------------------------
I have approached to some extent. Here is what I've done:
If G is a connected graph then we can easily find its spanning tree and then removing any two leaves x and y from the tree one at a time gives us 2 another graphs G − x and G − y such that they are both connected.
"Hence any connected graph G is a good graph."
1. I'm stuck on how to solve part (i).
2. I have designed the algorithm for the part(ii) which extracts the spanning tree from G and then finds two such leaves x and y. Clearly
it is linear in terms of #vertices(=n) & #edges(=m) i.e. O(n+m)
3. I have also solved the third part with a graph as follows:
O------------------O-----------------O
In the above graph G we denote three vertices as u1, u2, u3. The edges are (u1,u2) and (u2,u3). Clearly G − u2 is not connected.
Please help me in solving part(i).
Thanks in advance :)