Trees with distinguishing index equal distinguishing number plus one
Saeid Alikhani, Sandi Klav\v{z}ar, Florian Lehner, Samaneh Soltani

TL;DR
This paper characterizes trees where the distinguishing index equals the distinguishing number plus one and shows that for connected unicyclic graphs, these two parameters are equal.
Contribution
It provides a characterization of trees with the maximum difference between the distinguishing index and number, and establishes equality for connected unicyclic graphs.
Findings
Trees with $D'(G) = D(G) + 1$ are characterized.
For connected unicyclic graphs, $D'(G) = D(G)$ holds.
The inequality $D'(G) leq D(G) + 1$ is sharp for certain trees.
Abstract
The distinguishing number (index) () of a graph is the least integer such that has an vertex (edge) labeling with labels that is preserved only by the trivial automorphism. It is known that for every graph we have . In this note we characterize trees for which this inequality is sharp. We also show that if is a connected unicyclic graph, then .
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · Graph theory and applications
