Graphs and Combinatorics

, 25:489 | Cite as

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(n4) 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

Copyright information

© Springer 2009

Authors and Affiliations

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

Personalised recommendations