On fixing and distinguishing numbers of trees

TL;DR

This paper investigates the properties of distinguishing and fixing numbers in trees, providing bounds, characterizations, and universal constructions for trees with specific automorphism-distinguishing properties.

Contribution

It introduces new bounds on fixing numbers for distinguishable trees, characterizes trees with given radius and distinguishability, and constructs universal trees for these properties.

Findings

Fixing number of 2-distinguishable trees is at most 4n/11.

Universal trees characterize D-distinguishable trees by their branches.

Bounds relate distinguishing and fixing numbers to vertex eccentricities.

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.