# What is difference between cycle, path and circuit in Graph Theory

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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I think it is because various books use various terms differently. What some call a path is what others call a simple path. Those who call it a simple path use the word walk for a path. The same is true with Cycle and circuit. So, I believe that both of you are saying the same thing. What about the length? Some define a cycle, a circuit or a closed walk to be of nonzero length and some do not mention any restriction. A sequence of vertices and edges... could it be empty? I guess things should be standardized in Graph theory. –  user206676 Jan 9 at 8:47

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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cycle is a closed path with no vertices repeated.

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Yeah, but then what is a path and what is a circuit? –  Mike Pierce Feb 3 at 6:31

Different books have different terminology in some books a simple path means in which none of the edges are repeated and a circuit is a path which begins and ends at same vertex,and circuit and cycle are same thing in these books.

Books which use the term walk have different definitions of path and circuit,here, walk is defined to be an alternating sequence of vertices and edges of a graph, a trail is used to denote a walk that has no repeated edge here a path is a trail with no repeated vertices, closed walk is walk that starts and ends with same vertex and a circuit is a closed trail.

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