The $k$-in-a-tree problem for graphs of girth at least~$k$

TL;DR

This paper presents an $O(n^4)$ algorithm to determine if a graph with girth at least $k$ contains an induced tree covering $k$ specified vertices, extending previous results for $k=3$ and $k=4$ to all $k \\geq 5$.

Contribution

It introduces a structural characterization of graphs with girth at least $k$ that lack such induced trees, enabling the algorithm for all $k \\geq 5$.

Findings

Algorithm runs in $O(n^4)$ time.

Extends known results for $k=3$ and $k=4$ to all $k \\geq 5$.

Provides a structural description of relevant graphs.

Abstract

For all integers $k\geq 3$, we give an $O(n^4)$ time algorithm for the problem whose instance is a graph $G$ of girth at least $k$ together with $k$ vertices and whose question is "Does $G$ contains an induced subgraph containing the $k$ vertices and isomorphic to a tree?". This directly follows for $k=3$ from the three-in-a-tree algorithm of Chudnovsky and Seymour and for $k=4$ from a result of Derhy, Picouleau and Trotignon. Here we solve the problem for $k\geq 5$. Our algorithm relies on a structural description of graphs of girth at least $k$ that do not contain an induced tree covering $k$ given vertices ($k\geq 5$).