图论-当一个节点的所有边都包含在一个循环中时

Question

Given a directed or undirected graph, there are many algorithms to find cycles. However, I am looking for a specific type of cycle. A cycle in which a node [0...n] in my graph, which has "k" outward edges, has all k of it's edges contained to that cycle. That is to say, all of the edges of our node only connect to other nodes that make up the cycle.

Example:

We have a cycle (in an undirected graph, this time): ABDECA

In my matrix, node B has the following edges:

BA, BE, BD

This cycle qualifies as the type of feature I am looking for.

It is not required that every node in the cycle meet this rule- it is only required that at least one node in the cycle meets the rule.

So my question is - is there a name for this type of feature? And, is there an algorithm or does anyone even have a suspicion that an algorithm could be made to find such features more efficiently than randomly searching for cycles and then checking if any of their nodes meet the rule?

Answer 1

You can find all the cycles in a component of a graph by doing a DFS and finding all back edges (eg here ). This is an O(n + m) operation.

In the worst case, you could build a hash table (set) of all the nodes in the cycle and just check the connections of every node against it. There is no additional cost to building the hash table in terms of complexity, since you can do it while you unwind the stack that created the cycle and insertions are O(1), but checking against it may be expensive since the number of edges per node is indeterminate. You can probably filter out a significant number of candidates immediately by marking their connections as you go. I can not think of a fast algorithm to do this at the moment, but I will revisit this question when I do.