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Consider the following problem: we are given an undirected graph $G=(V,E)$ and three terminal vertices $t_1,t_2,t_3\in V$. We are asked whether there exists a set of vertices $S\subseteq V$ such that the induced subgraph $G[S]$ is a tree that contains all the terminals. In other words, we want to select vertices of $G$ so that the terminals become connected, but without inducing any cycles.

I would probably call this problem INDUCED CONNECTING TREE, or INDUCED STEINER TREE, but I haven't been able to find any references to it. Is this problem known by another name? Note that I'm not asking to minimize the size of the tree, I only ask whether such a tree exists. The answer may be no, even in a connected graph (for example, if we take a $K_3$ and attach a leaf terminal to each vertex).

More broadly, I would be interested in the (parameterized) complexity of this problem for a fixed number $k$ of terminals, if any such results are known. The problem is trivial for $k=2$ (find a shortest path) and $k=3$ is the first case for which no algorithm seems obvious (to me). I think showing that the problem is NP-complete when $k$ is part of the input should not be too hard (but I would appreciate a reference if this is known!).

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    $\begingroup$ This problem is well-studied under the name "three-in-a-tree" and more generally "k-in-a-tree". $\endgroup$
    – Laakeri
    Jan 1, 2023 at 21:47
  • $\begingroup$ @Laakeri Thanks, this is the name I was looking for. Apparently, the complexity is open for fixed $k\ge 4$, but the problem is in P for $k=3$. If you turn your comment into an answer, I will be happy to accept it. $\endgroup$ Jan 2, 2023 at 10:40

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This problem is well-studied under the name "three-in-a-tree" and more generally "k-in-a-tree". A polynomial-time algorithm for three-in-a-tree is given in [1], but the complexity of $k$-in-a-tree seems to be open for every fixed constant $k \ge 4$.

[1] Chudnovsky, Maria, and Paul Seymour. "The three-in-a-tree problem." Combinatorica 30.4 (2010): 387-417.

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