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Computing fixed point free automorphisms of graphs

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

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.

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.

Paper Structure

This paper contains 17 sections, 31 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hardness reductions
  3. Algorithm for FPFAut
  4. Modular Decomposition
  5. PFPFAut
  6. Algorithm for FPFAut
  7. Complexity
  8. Positive results
  9. Modular-width
  10. Tree-cographs
  11. $P_4$-sparse graphs
  12. Fixed point free involutions
  13. NP-Hard classes and fixed edge free involutions
  14. Algorithm for FPFInv
  15. Equitable partitions
  16. ...and 2 more sections

Key Result

Theorem 1.1

FPFAut is NP-hard for split, bipartite, $k$-subdivided, and $H$-free graphs, if $H$ is not an induced subgraph of $P_4$.

Figures (2)

  • Figure 1: a) Base graph $G = P_4$. b) Split graph construction. c) Bipartite construction. d) Subdivision construction.
  • Figure 2: Cyclops spider with $|K|=|S|=8$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.5
  • proof
  • Theorem 2.7
  • proof
  • Theorem 2.8
  • ...and 44 more