Quantum partial automorphisms of finite graphs

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