Graphs can contain circuits, i.e. paths which start and end at the same node. Most circuits are 'non-flat', like this:

Graph with circuit

I'm trying to find a term that describes a particular type of circuit. You might describe it as 'flat', 'zero-volume' or 'doubling-back'.

The circuit visits some nodes in the following order: $[n_{1}, n_{2}, \dots, n_{x-1}, n_{x}, n_{x-1}, \dots, n_{2}, n{1}]$.

Here's an illustration of the same concept:

Graph with 'flat' circuit

  • $\begingroup$ A simple graph has undirected edges, and only a single edge between any two vertices. So, among simple graphs, any path could be deemed a 'flat' circuit, should you so choose. A directed graph similarly tends to have a single edge between any two vertices. But, that edge may be unidirectional or bidirectional. If every edge is bidirectional, then every path yields a 'flat' circuit. So, it is not entirely clear what you are looking for. Depending on the context, there could be more than one name for what you want. $\endgroup$ Apr 22, 2020 at 12:22

1 Answer 1


The terminology I see most commonly used is as follows: A closed walk is a sequence of adjacent vertices which starts and ends at the same vertex (your second example is a closed walk). A cycle is a closed walk with no repeated edges or vertices, except the start and end vertex (your first example is a cycle).

So your 'flat circuits' would be referred to as 'closed walks with no induced cycle'.


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