GATE | GATE CS 2008 | Question 85

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

Answer: (D)

Explanation: 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.

Thanks to Renjith P for providing above explanation.

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