Panchromatic patterns by paths

Let $H=(V_H,A_H)$ be a digraph, possibly with loops, and let $D=(V_D, A_D)$ loopless multidigraph colouring of its arcs $c: A_D \rightarrow V_H$. An $H$-path $D$ is path $(v_0, \dots, v_n)$ such that $(c(v_{i-1}, v_i), c(v_i,v_{i+1}))$ an arc $H$ for every $1 \le i n-1$. For $u, v \in V_D$, we say $u$ reaches $v$ by $H$-paths if there exists from to in $D$. A subset $S \subseteq V_D$ $H$-absorbent vertex $V_D-S$ some $S$, it $H$-independent no $S$ can reach another (different) $H$-pahts. $H$-kernel independent absorbent $V_D$. We define $\tilde{\mathscr{B}}_1$ as the set digraphs any $H$-arc-coloured tournament has paths vertex; $\tilde{\mathscr{B}}_2$ consists digraph independent, set; analogously, $\tilde{\mathscr{B}}_3$ contains paths. In this work, present characterization $\tilde{\mathscr{B}}_2$, provide structural properties which settle up except analysis single on three vertices.