Homotopy groups of Hom complexes of graphs

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 ${\text{Hom}}_{\ast }(G,H)$ with the homotopy groups of ${\text{Hom}}_{\ast }(G,H^I)$. Here ${\text{Hom}}_{\ast }(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $\text{Hom}$ 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$\Omega H$isthegraphofbasedclosedpathsin$H$and$[G,K]{\times}$isthesetof$\times$-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.