Discussion
Let G be a connected graph with at least 3 vertices such that removal of any edge of G disconnects the graph. Then which of the following is not true ?
Answer: D
The statement that is not true is G has at least two spanning trees.
A connected graph in which removing any edge disconnects it has no cycle (an edge on a cycle could be removed without disconnecting), so G is a tree. Then:
- G has no circuit (true).
- There is a unique path between any two vertices (true for trees).
- A tree with at least 2 vertices has at least two pendant vertices (true).
But a spanning tree of a tree must use all of its \(n - 1\) edges, so the only spanning tree of G is G itself. G has exactly one spanning tree, not two.
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