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Consider the following statements:

1. A graph in which there is a unique path between every pair of vertices is a tree.
2. A connected graph with e=v-1 is a tree
3. A connected graph with e=v-1 that has no circuit is a tree

Which one of the above statements is/are true?

1. I and III
2. II and III
3. I and II
4. All of the above
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Lets go one by one,

(i) A graph in which there is a unique path between every pair of vertices is a tree.

==> This statement is true, Because graph can have unique path only when it does not have cycle. And according to the definition of tree, its a graph without cycel. Hence this is a valid statement.

(ii) A connected graph with e = v − 1 is a tree.

==> This statement is true. Not every graph with e = v - 1, will be a tree. But if the graph is connected hence its true.

(iii) A graph with e = v − 1 that has no circuit is a tree.

==> This statement is also true, Here he has not mentioned the connected thing, but mentioned that it has no circuit. It means that its connected.

You can also proof these, just by taking some example. For the formal Proof check this

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i need proof.
check now.
u r a savier sir thanks
What is meaning of circuit here ??

I know two types of circuit - A ) Hamilton circuit and B) Eulerian Circuits.

Circuit means cycle. and they are not two types of circuit.