On The Fixatic Number of Graphs

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.