Abstract :
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.
