$(\K,\leq)$ has the amalgamation property for strong extensions.

For any finite $L_n$-graphs $A\subseteq B,C$ we denote by $B\otimes_AC$ the *free amalgam* of $B$ and $C$ over $A$, i.e. the graph on $B\cup C$ containing no edges between elements of $B\setminus A$ and $C\setminus A$.

If $A\leq B,C$ are in $\K_n$, then $D:=B\otimes_AC\in\K$ and $B,C\leq D$.

Clearly, $B,C\leq D$. To see that $D\in\K_n$, note that if $B\in \K_n$ and $B'$ is an elementary strong extension of $B$, then also $B'\in\K_n$. This is clear for strong extensions of type 1. For strong extensions of type 2. suppose that $(b,a,b')$ is a path with $a\in V_i,b,b'\in V_{i-1}$, and $\gamma=(a,b,\ldots , b',a)\subset B'$ is an $E_i\cup\ldots\cup E_{i-j}$-path of length $k$ containing the new vertex $y$. Since the new vertex has exactly two neighbours $y_1,y_2$, this implies that the vertex is of type $V_m$ for some $i-j\leq m\leq i$ and $(y_1,y,y_2)$ is contained in $\gamma$. By construction of strong extensions, there is some $z\in B$ such that $(y_1,z,y_2)$ is a path. Hence we may replace all occurrences of $y$ in $\gamma$ by $z$. Then $\gamma$ is contained in $B$ and we find the required path in $R(a)$ with all $V_{i-j}$-vertices occurring in $\gamma$.

This shows 