The four-in-a-tree problem in triangle-free graphs

TL;DR

This paper characterizes the structure of triangle-free graphs lacking an induced tree covering four vertices and provides an efficient algorithm to find such a tree or certify its absence, along with complexity results.

Contribution

It offers a structural characterization of certain triangle-free graphs and introduces an $O(nm)$-time algorithm for the four-in-a-tree problem in this class, plus complexity insights.

Findings

Graphs without a four-vertex induced tree resemble squares or cubes.

Provided an efficient algorithm to find or certify the absence of such trees.

Proved NP-completeness for a related tree covering problem.

Abstract

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time $O(n^4)$ whether three given vertices of a graph belong to an induced tree. Here, we study four-in-a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the "same structure", in a sense to be defined precisely, as a square or a cube. We provide an $O(nm)$-time algorithm that given a triangle-free graph $G$ together with four vertices outputs either an induced tree that contains them or a partition of $V(G)$ certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree $T$ covering the four vertices such that at most one vertex of $T$ has degree at least 3 is NP-complete.