Computing fixed point free automorphisms of graphs
Abstract
In 1981, Lubiw proved that the fixed point free automorphism problem (FPFAut) is NP-complete: given a graph G, determine whether there exists an automorphism that maps no vertex of G to itself. We revisit this problem and prove that FPFAut remains NP-complete when restricted to split, bipartite, k-subdivided, and H-free graphs, if H is not an induced subgraph of P_4. The class of P_4-free graphs receives the special name of cographs. We provide a polynomial time algorithm for three extensions of cographs: bounded modular-width graphs, tree-cographs and P_4-sparse graphs. Our approach uses the well known modular decomposition of graphs. As a consequence, we generalize a result of Abiad et. al. on the problem of computing 2-homogeneous equitable partitions.
Cite
@article{arxiv.2603.26006,
title = {Computing fixed point free automorphisms of graphs},
author = {Aida Abiad and Gabriel Coutinho and Emanuel Juliano and Vinicius F. dos Santos and Sjanne Zeijlemaker},
journal= {arXiv preprint arXiv:2603.26006},
year = {2026}
}
Comments
20 pages, 2 figures