Computing fixed point free automorphisms of graphs

2026-03-26

AI Summary

This study addresses the Fixed-Point-Free Automorphism (FPFAut) problem, which asks whether a graph admits an automorphism that moves every vertex. By integrating modular decomposition with structural graph analysis, the authors establish that FPFAut is NP-complete on split graphs, bipartite graphs, k-subdivision graphs, and H-free graphs excluding the induced path P₄. In contrast, they present the first polynomial-time algorithms for solving FPFAut on graphs of bounded modular width, co-graphs, and P₄-sparse graphs. These results not only extend the tractability frontier to three new classes of P₄-free graphs but also generalize existing theory concerning 2-homogeneous equitable partitions.

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.

Problem

• fixed point free automorphism
• graph automorphism
• NP-complete
• cographs
• modular decomposition

Innovation

• fixed point free automorphism
• modular decomposition
• cographs
• NP-completeness
• polynomial-time algorithm