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I am currently studying Graph Theory and want to know the difference in between Path , Cycle and Circuit.

I know the difference between Path and the cycle but What is the Circuit actually mean.

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Usually a path in general is same as a walk which is just a sequence of vertices such that adjacent vertices are connected by edges. Think of it as just traveling around a graph along the edges with no restrictions.

Some books, however, refer to a path as a "simple" path. In that case when we say a path we mean that no vertices are repeated. We do not travel to the same vertex twice (or more).

A cycle is a closed path. That is, we start and end at the same vertex. In the middle, we do not travel to any vertex twice.

It will be convenient to define trails before moving on to circuits. Trails refer to a walk where no edge is repeated. (Observe the difference between a trail and a simple path)

Circuits refer to the closed trails, meaning we start and end at the same vertex.

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In a circuit we have can repeated vertices, but we cannot in a cycle.

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I think I disagree with Kelvin Soh a bit, in that he seems to allow a path to repeat the same vertex, and I think this is not a common definition. I would say:

Path: Distinct vertices $v_1,\dots,v_k$ with edges between $v_i$ and $v_{i+1}$, $1 \le i \le k-1$.

Trail: A sequence of not necessarily distinct vertices $v_1,\dots,v_k$ and a sequence of edges $e_1,\dots,e_{k-1}$ such that $e_i$ connects $v_i$ and $v_{i+1}$, $1 \le i \le k-1$ and all of the $e_i$ are distinct.

Cycle: Distinct vertices $v_1,\dots, v_k$ with edges between $v_i$ and $v_{i+1}$, $1 \le i \le k-1$, and the edge $\{v_1,v_k\}$.

Circuit: A trail with the same first and last vertex.

Note: In some old texts the word circuit is sometimes used to mean cycle.

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