# The Four-in-a-Tree Problem in Triangle-Free Graphs

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## 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.

## Keywords

Tree Algorithm Three-in-a-tree Four-in-a-tree Triangle-free graphs Induced subgraph## AMS Mathematics Subject Classification

05C75 05C85 05C05 68R10 90C35## References

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