Computing fixed point free automorphisms of graphs

TL;DR

This paper investigates the computational complexity of fixed point free automorphisms in various graph classes, proving NP-completeness in many cases and providing polynomial algorithms for specific graph subclasses.

Contribution

It extends NP-completeness results to broader graph classes and introduces polynomial algorithms for certain graph subclasses using modular decomposition.

Findings

FPFAut is NP-complete for split, bipartite, k-subdivided, and H-free graphs (not induced subgraph of P_4).

Polynomial algorithms are provided for bounded modular-width graphs, tree-cographs, and P_4-sparse graphs.

Generalizes a result on 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.