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

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