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G is a graph on n vertices and 2n - 2 edges. The edges of G can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for G?
  • a)
    For every subset of k vertices, the induced subgraph has at most 2k-2 edges
  • b)
    The minimum cut in G has at least two edges
  • c)
    There are two edge-disjoint paths between every pair to vertices
  • d)
    There are two vertex-disjoint paths between every pair of vertices
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
G is a graph on n vertices and 2n - 2 edges. The edges of G can be par...
Counter for option D is as follows. Take two copies of K4(complete graph on 4 vertices), G1 and G2. Let V(G1)={1,2,3,4} and V(G2)={5,6,7,8}. Construct a new graph G3 by using these two graphs G1 and G2 by merging at a vertex, say merge (4,5). The resultant graph is two edge connected, and of minimum degree 2 but there exist a cut vertex, the merged vertex.
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G is a graph on n vertices and 2n - 2 edges. The edges of G can be par...
Given:
- G is a graph on n vertices and 2n - 2 edges.
- The edges of G can be partitioned into two edge-disjoint spanning trees.

To find:
Which of the following is NOT true for G?

Solution:
Let's analyze each option and determine if it is true or false for graph G.

a) For every subset of k vertices, the induced subgraph has at most 2k-2 edges:
This statement is true. In any graph, the number of edges in the induced subgraph formed by a subset of vertices is at most the number of edges in the original graph. Since G has 2n - 2 edges, any induced subgraph formed by a subset of k vertices will have at most 2k - 2 edges.

b) The minimum cut in G has at least two edges:
This statement is true. In a graph with two edge-disjoint spanning trees, the minimum cut is formed by removing the edges that connect the two trees. Since G has two edge-disjoint spanning trees, the minimum cut will have at least two edges.

c) There are two edge-disjoint paths between every pair of vertices:
This statement is true. In a graph with two edge-disjoint spanning trees, there will be two edge-disjoint paths between every pair of vertices. This is because each spanning tree provides a unique path between any two vertices.

d) There are two vertex-disjoint paths between every pair of vertices:
This statement is NOT true. While it is true that G has two edge-disjoint spanning trees, it is not guaranteed that there will be two vertex-disjoint paths between every pair of vertices. It is possible for the two spanning trees to share some vertices, resulting in only one vertex-disjoint path between certain pairs of vertices.

Therefore, the correct answer is option 'D'.
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G is a graph on n vertices and 2n - 2 edges. The edges of G can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for G?a)For every subset of k vertices, the induced subgraph has at most 2k-2 edgesb)The minimum cut in G has at least two edgesc)There are two edge-disjoint paths between every pair to verticesd)There are two vertex-disjoint paths between every pair of verticesCorrect answer is option 'D'. Can you explain this answer?
Question Description
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