Problem 1170. Count the Number of Undirected Cycles in a Graph

Created by Joseph Kirk

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Given a symmetric adjacency matrix, determine the number of unique undirected cycles.
For example, the graph represented by adjacency matrix
A = [
  0 1 1 0 1
  1 0 1 1 0
  1 1 0 1 1
  0 1 1 0 0
  1 0 1 0 0];
has 6 cycles. They are:
[1 -> 2 -> 3 -> 1]
[1 -> 3 -> 5 -> 1]
[2 -> 3 -> 4 -> 2]
[1 -> 2 -> 4 -> 3 -> 1]
[1 -> 2 -> 3 -> 5 -> 1]
[1 -> 2 -> 4 -> 3 -> 5 -> 1]
The input is an adjacency matrix of 0s and 1s, and the output should be the number of unique (simple) undirected cycles in the graph.

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Last Solution submitted on Nov 22, 2025

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Alfonso Nieto-Castanon on 4 Jan 2013

could you please clarify, in your example, why cycles like '1->2->1', '1->5->1' or '3->5->3' are not counted?

Alfonso Nieto-Castanon on 4 Jan 2013

it seems the expected solution only counts cycles that include three or more nodes... is this correct?

Joseph Kirk on 5 Jan 2013

That is correct. In a directed graph, the cycle 1 -> 2 -> 1 actually uses two different edges, while in an undirected graph, the edge 1 -> 2 and 2 -> 1 are one and the same. I am only considering cycles valid if each node along the way is visited once and if edges are not used more than one time. I chose this criteria so that the number of cycles counted would match the results for the table provided here: http://mathworld.wolfram.com/GraphCycle.html

Alfonso Nieto-Castanon on 5 Jan 2013

got it. Thanks for the clarification!

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Problem 1170. Count the Number of Undirected Cycles in a Graph

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Problem 1170. Count the Number of Undirected Cycles in a Graph

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