On Fixing number of Functigraphs

The fixing number of a graph $G$ is the order of the smallest subset $S$ of its vertex set $V(G)$ such that stabilizer of $S$ in $G$, $\Gamma_{S}(G)$ is trivial. Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:v=g(u)\}$. In this paper, we study the behavior of the fixing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also study the fixing number of functigraphs of some well known families of graphs like complete graphs, trees and join graphs.
Comment: 10 pages, 1figure