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arXiv · 2603.26006

Computing fixed point free automorphisms of graphs

Aida Abiad·Gabriel Coutinho·Emanuel Juliano·Vinicius F. dos Santos·Sjanne Zeijlemaker
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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.

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Aida Abiad, Gabriel Coutinho, Emanuel Juliano, Vinicius F. dos Santos, Sjanne Zeijlemaker. 2026-03-27. Computing fixed point free automorphisms of graphs. https://arxiv.org/abs/2603.26006

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