  The partial automorphisms of a graph $X$ having $N$ vertices are the
bijections $\sigma:I\to J$ with $I,J\subset\{1,\ldots,N\}$ which leave
invariant the edges. These bijections form a semigroup $\widetilde{G}(X)$,
which contains the automorphism group $G(X)$. We discuss here the quantum
analogue of this construction, with a definition and basic theory for the
quantum semigroup of quantum partial automorphisms $\widetilde{G}^+(X)$, which
contains both $G(X)$, and the quantum automorphism group $G^+(X)$. We comment
as well on the case $N=\infty$, which is of particular interest, due to the
fact that $\widetilde{G}^+(X)$ is well-defined, while its subgroup $G^+(X)$,
not necessarily, at least with the currently known methods.
