Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) → V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 ) ∪ V ( G 2 ) and the edge set E ( G 1 ) ∪ E ( G 2 ) ∪ { u v : g ( u ) = 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.
