Question

A path is ____________ if the first and last vertices in the sequence are the same.

closed

cycle

open

final

Answer 1

A path is a cycle if the first and last vertices in the sequence are same.

A cycle is a graph in which only one vertex is repeated and i.e first and last vertices. A cycle has length at least 1. it means only one vertex is there which is first also and last also. A cycle is a simple path with the same vertex as the first and the last vertex in the sequence if the path is simple. Length of a cycle is calculated according to the number of edges in the cycle. Degree of each vertex is 2

closed path : In a closed path, the first and last vertex is repeated like cycle but both the edges and vertices may repeat in a closed path.

Open Path :

in the open path of a graph the first and last vertex is different.

Answer 2

Answer:

A path is $\boxed{{\displaystyle \text{closed}}}$ if the first and last vertices in the sequence are the same.

Explanation:

• Closed Path: Starts and ends at the same vertex (e.g., a loop: $A\to B\to C\to A$).

• Cycle: A special type of closed path with no repeated vertices/edges (except start/end).

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• Open Path: Starts and ends at different vertices.

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• "Final" is not a standard graph theory term.

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Example:

• Closed but not a cycle: $A\to B\to C\to D\to A\to B\to A$ (repeats edges/vertices).

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• Cycle: $A\to B\to C\to A$ (no repeats).