On Zero Forcing Number of Functigraphs

\emph{Zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the “AIM Minimum Rank – Special Graphs Work Group”. 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 $E=E(G_1)\cup E(G_2)\cup \{uv\mid v=f(u)\}$. For a connected graph $G$ of order $n\ge 3$, it is readily seen that $1+\delta(G) \le Z(C(G, \sigma)) \le n$foranypermutation$\sigma$;weshowthat$1+ \delta(G) \le Z(C(G, f)) \le 2n-2$foranyfunction$f$,where$\delta(G)$ is the minimum degree of $G$. We give examples showing that there does not exist a function $g$ such that, for every pair $(G,f)$, $Z(G)<g(Z(C(G,f)))$ or $g(Z(G))>Z(C(G,f))$. We further investigate the zero forcing number of functigraphs on complete graphs, on cycles, and on paths.