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On fixing and distinguishing numbers of trees

Calum Buchanan, Peter Dankelmann, Isabel Harris, Paul Horn, K. E. Perry, Emily Rivett-Carnac

Abstract

A graph $G$ is $D$-distinguishable if there is a labeling of its vertices with $D$ labels such that the only automorphism of $G$ which preserves the labeling is the identity. The distinguishing number of $G$ is the minimum value $D$ for which $G$ is $D$-distinguishable. The fixing number of $G$ is the minimum cardinality of a subset of the vertices of $G$ which is fixed pointwise only by the trivial automorphism. We prove that the fixing number of any $2$-distinguishable tree of order $n \geq 3$ is at most $4n/11$, or at most $(D-1)n / (D+1)$ for a $D$-distinguishable tree ($D \geq 3$). For every $D$ and $r$ at least $2$, we characterize the $D$-distinguishable trees with radius $r$ by constructing a universal tree $T_r^D$ which has the property that a tree $T$ of radius $r$ is $D$-distinguishable if and only if $T$ is a union of branches of $T_r^D$. We obtain a similar collection of universal trees for the property of having a constant paint cost spectrum, i.e., the minimum size of the complement of a color class in a distinguishing $D$-coloring of $T$ is equal to the fixing number. Finally, we prove bounds on the distinguishing and fixing numbers of a tree in terms of the eccentricities of its vertices.

On fixing and distinguishing numbers of trees

Abstract

A graph is -distinguishable if there is a labeling of its vertices with labels such that the only automorphism of which preserves the labeling is the identity. The distinguishing number of is the minimum value for which is -distinguishable. The fixing number of is the minimum cardinality of a subset of the vertices of which is fixed pointwise only by the trivial automorphism. We prove that the fixing number of any -distinguishable tree of order is at most , or at most for a -distinguishable tree (). For every and at least , we characterize the -distinguishable trees with radius by constructing a universal tree which has the property that a tree of radius is -distinguishable if and only if is a union of branches of . We obtain a similar collection of universal trees for the property of having a constant paint cost spectrum, i.e., the minimum size of the complement of a color class in a distinguishing -coloring of is equal to the fixing number. Finally, we prove bounds on the distinguishing and fixing numbers of a tree in terms of the eccentricities of its vertices.
Paper Structure (7 sections, 16 theorems, 23 equations, 7 figures)

This paper contains 7 sections, 16 theorems, 23 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Fixing densities of $D$-distinguishable trees
  4. Universal $D$-distinguishable trees
  5. Eccentric sequences
  6. Future Work
  7. Acknowledgements

Key Result

Theorem 1

For every tree $T$, there is a minimum cardinality fixing set consisting only of leaves of $T$.

Figures (7)

  • Figure 1: Colorings of the cycle of length $6$ which minimize the $2$- and $3$-paint cost, respectively.
  • Figure 2: A $2$-distinguishable tree with fixing density $4/11$.
  • Figure 3: Universal D-distinguishable Trees with $r = 2$.
  • Figure 4: Universal $D$-distinguishable tree for the property $\rho^{D(T)}(T) = F(T)$
  • Figure 5: The exceptions to the bounds in Theorem \ref{['theo:fixing-distinguishing-vs-eccentric-sequence']}
  • ...and 2 more figures

Theorems & Definitions (30)

  • Theorem 1: EH2006
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark
  • Theorem 5
  • proof
  • Lemma 6
  • ...and 20 more