  An oriented graph is a digraph that does not contain a directed cycle of
length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as
an induced sub(di)graph. The Gy\'arf\'as-Sumner conjecture is a widely-open
conjecture on simple graphs, which states that for any forest $F$, there is
some function $f$ such that every $F$-free graph $G$ with clique number
$\omega(G)$ has chromatic number at most $f(\omega(G))$. Aboulker, Charbit, and
Naserasr [Extension of Gy\'arf\'as-Sumner Conjecture to Digraphs; E-JC 2021]
proposed an analogue of this conjecture to the dichromatic number of oriented
graphs. The dichromatic number of a digraph $D$ is the minimum number of colors
required to color the vertex set of $D$ so that no directed cycle in $D$ is
monochromatic.
  Aboulker, Charbit, and Naserasr's $\overrightarrow{\chi}$-boundedness
conjecture states that for every oriented forest $F$, there is some function
$f$ such that every $F$-free oriented graph $D$ has dichromatic number at most
$f(\omega(D))$, where $\omega(D)$ is the size of a maximum clique in the graph
underlying $D$. In this paper, we perform the first step towards proving
Aboulker, Charbit, and Naserasr's $\overrightarrow{\chi}$-boundedness
conjecture by showing that it holds when $F$ is any orientation of a path on
four vertices.
