The cost of symmetry in connected graphs

TL;DR

This paper investigates the bounds on the cost of symmetry in connected graphs, demonstrating that the previously established estimate is not always optimal by providing a counterexample.

Contribution

It proves that the known estimate for the cost of symmetry in graphs is not universally optimal by constructing a specific counterexample graph.

Findings

The original estimate for the cost of symmetry is not always tight.

A specific graph is constructed to show the estimate's non-optimality.

Abstract

The paper answers the question posed in a joint paper by A. A. Klyachko and N. M. Luneva about the optimality of the estimate for the cost of symmetry in graphs. The original estimate says that if n vertices can be removed from a connected graph so that there is no connected subgraph of isomorphic $\Gamma$ left in it, then at most $n|V(\Gamma)|$ vertices that form a set invariant under all automorphisms of the graph so that the graph does not contain a subgraph isomorphic to $\Gamma$. We will prove that there exists a graph $\Gamma$ for which this estimate is not optimal.