On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim (G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f:V(G_1)\to V(G_2)$ be a function. Then a \emph{functigraph} $C(G,f)=(V,E)$ has the vertex set $V=V(G_1)\cup V(G_2)$ and the edge set $E=E(G_1)\cup E(G_2)\cup \{uv\mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2\le \dim (C(G,f))\le 2n-3$, if $G$ is a connected graph of order $n\ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.