On retracts, absolute retracts, and folds in cographs

TL;DR

This paper investigates the computational complexity of the retract problem in cographs, showing NP-completeness generally, but tractability in special subclasses and fixed-parameter cases, and characterizes absolute retracts.

Contribution

It establishes NP-completeness of the retract problem in cographs and identifies conditions for polynomial-time solutions and fixed-parameter tractability, also characterizing absolute retracts.

Findings

Retract problem is NP-complete for cographs

Problem is fixed-parameter tractable by H's size

Polynomial-time solutions for threshold and trivially perfect graphs

Abstract

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem 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 problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.