# On The Fixatic Number of Graphs

## TL;DR

This paper introduces and studies the fixatic number of graphs, providing bounds, exact values under certain conditions, and realizability results, thereby advancing understanding of graph automorphisms and fixing sets.

## Contribution

It defines the fixatic number, establishes bounds and exact values, and explores realizability, contributing new theoretical insights into graph automorphism fixing sets.

## Key findings

- Established sharp bounds for the fixatic number.
- Derived exact fixatic numbers under specific conditions.
- Presented realizability results for the fixatic number.

## Abstract

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $\Pi$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition of $V(G)$. Then $\Pi$ is called a {\it fixatic partition} if for all $i$; $1\leq i\leq k$, $F_i$ is a fixing set for $G$. The cardinality of a largest fixatic partition is called the {\it fixatic number} of $G$. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper.