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Let $G$ be a graph and let $v$ be a vertex. Is there a polynomial algorithm for the following operation?

Operation. Find all the induced paths in $G$ with first vertex $v$.


Background

This problem is the confusion I encountered while reading the article Induced paths in graphs without anticomplete cycles. The authors asserted that there is a polynomial algorithm to check whether a graph contains no two non-adjacent cycles (and more generally no k non-adjacent cycles). Their algorithm is as follows:

  • For each vertex $v$, find all the induced paths with first vertex $v$.
  • For each induced path $P$, find all induced cycles that consist of $P$ and one extra vertex.
  • Check whether any two of these cycles are disjoint and have no edges between them.

I am puzzled by the first step. If that's the case, then I think that finding the longest induced path in a graph would be possible with a polynomial algorithm. But as far as I know, it is NP-hard. See https://en.wikipedia.org/wiki/Induced_path

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    $\begingroup$ I think it is best to provide some background information on the problem and write down some of your thoughts. After all, the purpose of CS SE is to built a repository of meaningful information for future members. $\endgroup$ Feb 14 at 16:46
  • $\begingroup$ @codeing_monkey Thank you. I will write it though I think it has many words. $\endgroup$
    – licheng
    Feb 15 at 0:35

1 Answer 1

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Consider the following graph $G = (V, E)$ where:

  • $V = \{v_0\}\cup \{u_1,…, u_n\}\cup \{v_1, …, v_n\}$;
  • $E = \{\{v_0,u_1\}, \{v_0, v_1\}\}\cup \bigcup\limits_{k=1}^{n-1}\{\{u_k, u_{k+1}\}, \{u_k, v_{k+1}\}, \{v_k, u_{k+1}\}, \{v_k, v_{k+1}\}\}$.

enter image description here

Then the graph has more than $2^n$ induced paths from $v_0$. So there is no way to find all of them in polynomial time.

I think the paper states that there is a polynomial-time algorithm to find them if you make additionnal hypotheses on the graph (though I agree that many details are missing).

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  • $\begingroup$ I agree. But, I am surprised that the authors claim their polynomial algorithm (to check whether a graph contains no two non-adjacent cycles (and more generally no k non-adjacent cycles)) follows from the fact that these graphs have a polynomial number of induced paths. In fact, before checking, we don't know how many induced paths a given graph has. Note: The testing graph given is general. $\endgroup$
    – licheng
    Feb 15 at 17:34
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    $\begingroup$ Well that could still be true: you could enumerate all induced paths, and answer false when the number of induced paths becomes greater than $|G|^c$. However I am not an expert on the question. Perhaps you could contact the authors to ask further questions? $\endgroup$
    – Nathaniel
    Feb 15 at 17:38
  • $\begingroup$ I agree with you. Perhaps we need to determine what c actually is before this algorithm is implemented. I will ask the authors if they have implemented the algorithm. $\endgroup$
    – licheng
    Feb 15 at 17:51

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