Optimization Letters

, Volume 12, Issue 3, pp 535–549 | Cite as

On retracts, absolute retracts, and foldings in cographs

  • Ton Kloks
  • Yue-Li Wang
Original Paper


A retract of a graph G is an induced subgraph H of G such that there exists a homomorphism \(\rho :G \rightarrow H\). When both G and H are cographs, we show that the problem to determine whether H is a retract of G is NP-complete; moreover, we show that this problem on cographs is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the retract problem becomes tractable in polynomial time. The retract problem is also solvable in linear time when one cograph is given as an induced subgraph of the other. We characterize absolute retracts for the class of cographs. Foldings generalize retractions. We show that the problem to fold a trivially perfect graph onto a largest possible clique is NP-complete.


Retracts Absolute retracts Foldings Cographs 



The authors would like to thank anonymous referees for their careful reading with corrections and useful comments which helped to improve the paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan
  2. 2.Department of Information ManagementNational Taiwan University of Science and TechnologyTaipeiTaiwan

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