On the Distinguishing Number of Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph G and $g:V\left(G_1\right)\to V\left(G_2\right)$ be a function. A functigraph $F_G$ consists of the vertex set $V\left(G_1\right)\cup V\left(G_2\right)$ and the edge set $E\left(G_1\right)\cup E\left(G_2\right)\cup \{uv:g\left(u\right)=v\}$ . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to $F_G$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs. View Full-Text