  An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by
replacing each edge by exactly one of the two possible arcs with the same end
vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by
$d^-_D(v)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of
$G$ is proper if $d^-_D(u)\neq d^-_D(v)$, for all $uv\in E(G)$. An orientation
with maximum indegree at most $k$ is called a $k$-orientation. The proper
orientation number of $G$, denoted by $\overrightarrow{\chi}(G)$, is the
minimum integer $k$ such that $G$ admits a proper $k$-orientation. We prove
that determining whether $\overrightarrow{\chi}(G) \leq k$ is NP-complete for
chordal graphs of bounded diameter, but can be solved in linear-time in the
subclass of quasi-threshold graphs. When parameterizing by $k$, we argue that
this problem is FPT for chordal graphs and argue that no polynomial kernel
exists, unless $NP\subseteq coNP/\ poly$. We present a better kernel to the
subclass of split graphs and a linear kernel to the class of cobipartite
graphs.
  Concerning bounds, we prove tight upper bounds for subclasses of block
graphs. We also present new families of trees having proper orientation number
at most 2 and at most 3. Actually, we prove a general bound stating that any
graph $G$ having no adjacent vertices of degree at least $c+1$ have proper
orientation number at most $c$. This implies new classes of (outer)planar
graphs with bounded proper orientation number. We also prove that maximal
outerplanar graphs $G$ whose weak-dual is a path satisfy
$\overrightarrow{\chi}(G)\leq 13$. Finally, we present simple bounds to the
classes of chordal claw-free graphs and cographs.
