Domination in Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$beafunction\mathrm{.}Thena\mathit{\text{functigraph}}$C(G, f)=(V, E)$hasthevertexset$V=V(G_1) \cup V(G_2)$andtheedgeset$E=E(G_1) \cup E(G_2) \cup {uv \mid u \in V(G_1), v \in V(G_2), v=f(u)}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma (G)$ denote the domination number of $G$. It is readily seen that $\gamma (G)\le \gamma (C(G,f))\le 2\gamma (G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.