domination number of complements of functigraphs

Let g=(v, e) be a simple graph. a subset s subseteq v(g) is a dominating set of g if every vertex in v(g) s is adjacent to at least one vertex in s. the domination number of graph g, denoted by γ(g), is the minimum size of a dominating set of vertices v(g). let g1 and g2 be two disjoint copies of graph g and f:v(g1) → v(g2) be a function. then a functigraph g with function f is denoted by c(g, f), its vertices and edges are v(c(g, f))=v(g1) cup v(g2) and e(c(g, f))=e(g1) cup e(g2) cup {vu| v in v(g1) , u in v(g2), f(v)=u}, respectively. in this paper, we investigate domination number of complements of functigraphs. we show that for any connected graph g,  γ(overlinec(g, f)) = 3. also we provide conditions for the function f in some graphs such that γ(overline{c(g, f))=3. finally, we prove if g is a bipartite graph or a connected k regular graph of order n ≥ 4 for k ∈ {2, 3, 4 } and g notin {k3, k4, k5, h1, h2}, then γ(overlinec(g, f)) = 2.