  The notion of $\times$-homotopy from \cite{DocHom} is investigated in the
context of the category of pointed graphs. The main result is a long exact
sequence that relates the higher homotopy groups of the space $\Hom_*(G,H)$
with the homotopy groups of $\Hom_*(G,H^I)$. Here $\Hom_*(G,H)$ is a space
which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the
usual $\Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a
corollary it is shown that $\pi_i \big(\Hom_*(G,H) \big) \cong [G,\Omega^i
H]_{\times}$, where $\Omega H$ is the graph of based closed paths in $H$ and
$[G,K]_{\times}$ is the set of $\times$-homotopy classes of pointed graph maps
from $G$ to $K$. This is similar in spirit to the results of \cite{BBLL}, where
the authors seek a space whose homotopy groups encode a similarly defined
homotopy theory for graphs. The categorical connections to those constructions
are discussed.
