On Fixing number of Functigraphs

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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, Γₛ(G) is trivial. Let G₁ and G₂ be disjoint copies of a graph G, and let g:V(G₁)→ V(G₂) be a function. A functigraph FG consists of the vertex set V(G₁)∪ V(G₂) and the edge set E(G₁)∪ E(G₂)∪ {uv:v=g(u)}. In this paper, we study the behavior of the fixing number in passing from G to FG 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.