Talk:Cycle space
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Cycle space via cuts
[edit]- The elements of the cycle space can be characterized by the cuts: A set of edges is an element of the cycle space if and only if it meets every cut in a finite number of edges.
This can't be quite right, since in a finite graph any set of edges would qualify. AxelBoldt 21:11, 24 Feb 2004 (UTC)
fundamental cycles?
[edit]Article currently states:
- It is not necessary to use all cycles to generate the cycle space: if G is connected and any spanning tree T of G is given, then the fundamental cycles of T form a basis of the cycle space.
I don't really understand what this is saying. Clearly, adding just one edge to a spanning tree makes a cycle, and also its clear that one gets one unique cycle for every edge that is not in the spanning tree. Is this the definition of a "fundamental cycle"? linas (talk) 23:12, 7 September 2008 (UTC)
- Never mind, yes, this is the definition. linas (talk) 01:39, 8 September 2008 (UTC)
Homology
[edit]Shouldnt the article mention that if the graph is considered as a topological space the cycle space is natually identified with the first homology group?81.101.138.148 (talk) 10:02, 14 October 2011 (UTC) That was me Billlion (talk) 10:03, 14 October 2011 (UTC)
- Yes, that would be a good idea. —David Eppstein (talk) 14:38, 14 October 2011 (UTC)
Eulerian subgraphs and edge sets
[edit]The article often forgets to say "spanning" when mentioning Eulerian subgraphs.
I added the fact that the cycle space can be defined as containing edge sets of Eulerian subgraphs rather than Eulerian spanning subgraphs. This is a common interpretation of the cycle space. Zaslav (talk) 02:45, 21 January 2026 (UTC)