Locally arc-transitive graphs of valence $\{3,4\}$ with trivial edge kernel

In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\mathrm{\Gamma }(u)\cup \mathrm{\Gamma }(v)$ is trivial. We find nineteen finitely presented groups with the property that any such group $G$ is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence $3,4$ on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup $G$ with $G_{uv}^{[1]}=1$.