Characterization of graphs with distinguishing number equal list distinguishing number

TL;DR

This paper investigates the list distinguishing number of graphs, determines it for specific graph families, characterizes graphs where it equals the distinguishing number, and extends the characterization to other list parameters.

Contribution

It provides the first characterization of graphs where the distinguishing number equals the list distinguishing number, and extends this to other list-based graph parameters.

Findings

Determined list-distinguishing number for two graph families.

Characterized graphs with equal distinguishing and list distinguishing numbers.

Extended the characterization to other list parameters.

Abstract

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V (G)}$ of lists of labels to the vertices of $G$. A distinguishing $L$-labeling of $G$ is a distinguishing labeling of $G$ where the label of each vertex $v$ comes from $L(v)$. The list distinguishing number of $G$, $D_l(G)$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)| = k$ for all $v \in V (G)$ yields a distinguishing $L$-labeling of $G$. In this paper, we determine the list-distinguishing number for two families of graphs. We also characterize graphs with the distinguishing number equal the list distinguishing number. Finally, we show that this characterization works for other list numbers of a graph.