Internally connected graphs and the Kashiwara-Vergne Lie algebra

It is conjectured that the Kashiwara-Vergne Lie algebra ${\widehat{\mathfrak{k}\mathfrak{r}\mathfrak{v}}}_2$ is isomorphic to the direct sum of the Grothendieck-Teichm"uller Lie algebra ${\mathfrak{g}\mathfrak{r}\mathfrak{t}}_1$ and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of ${\widehat{\mathfrak{k}\mathfrak{r}\mathfrak{v}}}_2$ whose intersection is ${\mathfrak{g}\mathfrak{r}\mathfrak{t}}_1$, thus giving a way to interpolate between these two Lie algebras.