Graphs and Combinatorics

, 25:489

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

• Nicolas Derhy
• Christophe Picouleau
• Nicolas Trotignon
Article

## 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

1. 1.
Bienstock, D.: On the complexity of testing for odd holes and induced odd paths. Discrete Math., 90, 85–92, 1991 . See also Corrigendum by B. Reed, Discrete Math. 102, p. 109 (1992)Google Scholar
2. 2.
Chudnovsky, M., Seymour, P.: Excluding induced subgraphs. In: Surveys in Combinatorics, vol. 346, p. 99–119. London Mathematical Society Lecture Notes Series, 2007Google Scholar
3. 3.
Chudnovsky, M., Seymour, P.: The three-in-a-tree problem. Combinatorica (to appear)Google Scholar
4. 4.
Derhy, N., Picouleau, C.: Finding Induced Trees. Accepted for publication in Discrete Appl. Math., (2008)Google Scholar
5. 5.
Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, 1985Google Scholar
6. 6.
Lévêque, B., Lin, D., Maffray, F., Trotignon, N.: Detecting induced subgraphs. Manuscript, 2007Google Scholar

## Authors and Affiliations

• Nicolas Derhy
• 1
• Christophe Picouleau
• 1
• Nicolas Trotignon
• 2
1. 1.CEDRIC, CNAMParisFrance
2. 2.CNRS, LIAFAParis DiderotFrance