Corollary 3

Let $G'$ be a subgraph of G. The morphism from $\varpi_{G'}$ into $\varpi_G$ deduced from the injection of $G'$ into G is injective.

Let us denote by $i$ the inclusion morphism of $G'$ into G and by $\varpi (i)$ the morphism from $\varpi_{G'}$ into $\varpi_G$ deduced from it. The mappings Som($i$) and Som($\varpi (i)$) coincide. Moreover, if $c$ is a path without backtracking in $G'$, the path $i(c)$ is a path without backtracking in G. The assertion therefore results from Corollary 1.